The answer to the second question is also negative.  Here is a quick review of the Hartshorne-Serre construction in the smooth case.  Let $X$ be a smooth, connected, proper $k$-scheme of pure dimension $n\geq 2$.  Let $C\subset X$ be a codimension $2$, closed subscheme that is a local complete intersection scheme such that $H^0(C,\mathcal{O}_C)$ equals $k$ (this implies that $C$ is connected, and it is equivalent to connectedness if $C$ is reduced).

<B>Hypothesis 1.</B> There exists an invertible sheaf $\mathcal{L}$ on $X$ such that the $\mathcal{L}|_C$ is isomorphic to $\omega_{C/k}$ and such that $H^{n-2}(X,\mathcal{L})$ vanishes.

<B>Hartshorne-Serre Correspondence</B>.  Under the hypothesis, there exists a pair $(\mathcal{E},s)$ of a locally free $\mathcal{O}_X$-module $\mathcal{E}$ of rank $2$ and determinant $\omega^\vee_{X/k}\otimes_{\mathcal{O}_X} \mathcal{L}$ together with a global section $s$ whose zero scheme is precisely $C$.  Moreover, the set of isomorphism classes of such pairs is naturally a torsor for the $k$-vector space $H^1(X,\mathcal{L}^\vee\otimes_{\mathcal{O}_X}\omega_{X/k})$.

<B>Proof</B>.  Denote by $\mathcal{I}$ the ideal sheaf of $C$ in $X$.  Consider the Ext group $\text{Ext}^1_{\mathcal{O}_X}(\mathcal{I},\mathcal{L}^\vee\otimes_{\mathcal{O}_X}\omega_{X/k})$.  Since the left-exact functor $\text{Hom}_{\mathcal{O}_X}(\mathcal{I},-)$ is the composition of the sheaf Hom functor followed by the global sections functor, there is a corresponding Grothendieck spectral sequence computing the right derived functors.  The low terms of this spectral sequence give a long exact sequence.  

Before writing this long exact sequence, note first the canonical isomorphism,
$$
\textit{Ext}^1_{\mathcal{O}_X}(i_*\mathcal{O}_C,i_*\mathcal{O}_C) \cong i_*(i^*\mathcal{I})^\vee = i_* N_{C/X},
$$
where $i^*\mathcal{I}$ is the conormal sheaf of $C$ in $X$, and where $N_{C/X}$ denotes the dual locally free $\mathcal{O}_C$-module, i.e., the normal sheaf.  For every closed subscheme of arbitrary codimension $c$, equal to the rank of $N_{C/X}$, the exterior product on the Ext algebra defines a morphism of differential graded $i_*\mathcal{O}_C$-algebras,
$$
i_* \bigwedge^\bullet_{\mathcal{O}_C} N_{C/X} \to \textit{Ext}^\bullet_{\mathcal{O}_X}(i_*\mathcal{O}_C,i_*\mathcal{O}_C).
$$
Working locally, and using the fact that $C$ is a local complete intersection scheme, this morphism is an isomorphism.  Chasing exact sequences, $\textit{Ext}^r_{\mathcal{O}_X}(i_*\mathcal{O}_C,\mathcal{O}_X)$ is nonzero if and only if $r$ equals the codimension $c$, in which case it equals the pushforward of the determinant of $N_{C/X}$. Chasing long exact sequences of the short exact sequence relating $\mathcal{I}$ and $i_*\mathcal{O}_C$, we have
$$
\textit{Hom}_{\mathcal{O}_X}(\mathcal{I},\mathcal{F}) \cong \mathcal{F}, \ \ \textit{Ext}^{c-1}_{\mathcal{O}_X}(\mathcal{I},\mathcal{F}) \cong i_*(\text{det}(N_{C/X})\otimes_{\mathcal{O}_C} i^*\mathcal{F}),
$$
and all other sheaf Ext groups vanish.  In the case of interest, where $c$ equals $2$, this gives 
$$
\textit{Hom}_{\mathcal{O}_X}(\mathcal{I},\mathcal{L}^\vee\otimes_{\mathcal{O}_X}\omega_{X/k}) \cong \mathcal{L}^\vee\otimes_{\mathcal{O}_X}\omega_{X/k}, \ \ 
\textit{Ext}^1_{\mathcal{O}_X}(\mathcal{I},\mathcal{L}^\vee\otimes_{\mathcal{O}_X}\omega_{X/k}) \cong i_*\mathcal{O}_C.
$$
Now, the long exact sequence of the Grothendieck spectral sequence becomes
$$
0\to H^1(X,\mathcal{L}^\vee\otimes_{\mathcal{O}_X}\omega_{X/k}) \to \text{Ext}^1_{\mathcal{O}_X}(\mathcal{I},\mathcal{L}^\vee\otimes_{\mathcal{O}_X}\omega_{X/k}) \to H^0(C,\mathcal{O}_C) \xrightarrow{\delta} H^2(X,\mathcal{L}^\vee\otimes_{\mathcal{O}_X}\omega_{X/k}).
$$
By Serre duality, the last term is dual to $H^{n-2}(X,\mathcal{L})$.  This vanishes by hypothesis.  Thus, there exists an extension class that maps to the element $1$ in $H^0(C,\mathcal{O}_C)$, and the set of all such extension classes is a torsor under the $k$-vector space $H^1(X,\mathcal{L}^\vee\otimes_{\mathcal{O}_X}\omega_{X/k})$.  Interpreting the Ext group as a Yoneda Ext group, each such extension class is equivalent to a short exact sequence,
$$
0 \to \mathcal{L}^\vee\otimes_{\mathcal{O}_X}\omega_{X/k} \to \mathcal{E}' \to \mathcal{I} \to 0,
$$
such that the induced map 
$$
i^*\mathcal{E}' \to i^*\mathcal{I}
$$
is an isomorphism.  This is equivalent to local freeness of $\mathcal{E}'$ on a Zariski open neighborhood of $C$.  Since also $\mathcal{E}'$ is locally free on the complement of $C$, since also $\mathcal{I}$ is an invertible sheaf on the complement of $C$, these extension classes are precisely the ones for which $\mathcal{E}'$ is locally free of rank $2$.  The dual locally free sheaf $\mathcal{E}$ has a natural global section $s$ coming from $\textit{Hom}_{\mathcal{O}_X}(\mathcal{I},\mathcal{O}_X) \cong \mathcal{O}_X$.  The set of isomorphism classes of pairs $(\mathcal{E},s)$ is equivalent to the set of isomorphism classes of Yoneda extension classes above. <B>QED</B>

<B>Note</B>.  The history of the Hartshorne-Serre correspondence involves many other mathematicians: Horrocks, Barth, van de Ven, Grauert, M&uuml;lich, and Vogelaar.  I recommend the following article for more about this.

MR2351117 (2008g:14084) <br> 
Arrondo, Enrique <br>
A home-made Hartshorne-Serre correspondence. <br>
Rev. Mat. Complut. 20 (2007), no. 2, 423–443. <br>
https://pdfs.semanticscholar.org/982e/8d2ed4fcc0a0a445d5dbc81a361b368f8876.pdf

<B>Edit.</B> The following edit produces positive dimensional components of the moduli space of stable bundles, not just semistable bundles.  The original post is below the edit.

<B>Moduli spaces of rank $2$, stable bundles on a quintic threefold</B>. Now consider the special case that $X$ is a smooth quintic hypersurface in $\mathbb{P}^4$.  This is a Calabi-Yau manifold, i.e., $\omega_{X/k} \cong \mathcal{O}_X$.  By the Grothendieck-Lefschetz hyperplane theorem for Picard groups, every invertible sheaf on $X$ is isomorphic to $\mathcal{O}_{\mathbb{P}^4}(d)|_X$ for some integer $d$.  As a complete intersection, also $H^r(X,\mathcal{O}_{\mathbb{P}^4}(d)|_X)$ vanishes for $r=1$ and $r=2$.  Thus, for every complete intersection curve $C$ in $X$ with $H^0(C,\mathcal{O}_C)$ equal to $k$ and with $\omega_{C/k}$ equal to $\mathcal{L}|_C$ for some invertible sheaf $\mathcal{L}$ on $X$, the hypotheses of the Hartshorne-Serre correspondence are satisfied.  

<B>Hypothesis 1.</B> The restriction map $H^0(X,\mathcal{L})\to H^0(C,\mathcal{L}|_C)$ is surjective.

Under this hypothesis, also $H^1(X,\mathcal{L}\otimes \mathcal{I})$ vanishes.  Thus, associated to the defining short exact sequence, $$0\to \mathcal{O}_X \xrightarrow{s} \to \mathcal{E} \to \mathcal{L}\otimes\omega_{X/k}^\vee\otimes \mathcal{I} \to 0,$$ the long exact sequence of cohomology gives vanishing of $H^1(X,\mathcal{E})$.  Therefore, for every infinitesimal deformation of $\mathcal{E}$, there is a corresponding infinitesimal deformation of the global section $s$, and thus also an infinitesimal deformation of the zero scheme $C$ of $s$.  In other words, the morphism from the Hilbert scheme to the moduli space of sheaves is formally smooth.

<B>Hypothesis 3.</B> The invertible sheaf $\omega_C$ is isomorphic to $\mathcal{O}_{\mathbb{P}^4}(1)|_C$, i.e., $\mathcal{L}$ equals $\mathcal{O}_{\mathbb{P}^4}(1)|_X$.  Combined with Hypothesis 2, this is equivalent to the hypothesis that the embedding of $C$ in its linear span in $\mathbb{P}^4$ is a <b>canonical embedding</b>.

Under these hypotheses, from the defining short exact sequence above, it follows that $\mathcal{E}$ is stable.  The slope is $\mu=c_1(\mathcal{O}_{\mathbb{P}^4}(1))/2$.  An invertible sheaf $\mathcal{O}_{\mathbb{P}^4}(e)$ has slope $\geq \mu$ if and only if $e\geq 1$.  Since there is a nonzero morphism from $\mathcal{O}_{\mathbb{P}^4}(e-1)|_X$ to $\mathcal{I}$ if and only if $e-1 \leq -1$, i.e., if and only if $e\leq 0$, and similarly there is a nonzero morphism from $\mathcal{O}_{\mathbb{P}^4}(e)|_X$ to $\mathcal{O}_X$ if and only if $e\leq 0$, there is a nonzero morphism from $\mathcal{O}_{\mathbb{P}^4}(e)|_X$ to $\mathcal{E}$ if and only if $e\leq 0$.  Therefore $\mathcal{E}$ is stable.

It remains to describe the relevant family of canonical curves $C$.  For every line $R$ in $X$, for every $2$-plane $\Pi$ that contains $R$, the intersection $\Pi\cap X$ is a plane quintic $R\cup C$, where $C$ is a plane curve of degree $4$.  Thus, $C$ is a complete intersection curve with $H^0(C,\mathcal{O}_C)$ equal to $k$. Also $\omega_{C/k}$ equals $\mathcal{O}_{\mathbb{P}^4}(1)|_C$ by adjunction.  Thus, these residual curves satisfy the hypotheses above.  

Note that $H^0(X,\mathcal{E})$ equals $3$, i.e., there is a net of such curves associated to $E$.  This is consistent: there is also a net of $2$-planes $\Pi$ containing the line $R$ in $\mathbb{P}^4$.  

A general quintic threefold $X$ has precisely $2875$ lines, and thus $M$ would contain $2875$ isolated points.  However, there are many smooth quintic threefolds $X$ whose Fano scheme of lines has positive dimension, e.g., this holds if $X$ has a conical tangent hyperplane (the intersection of the tangent hyperplane with $X$ is a cone over a smooth plane quintic curve).  There exists such a quintic threefold with a conical point and with an isolated line.  For such a qunitic threefold, the moduli space $M$ of stable sheaves with fixed numerical invariants equal to those of $\mathcal{E}$ has one isolated component and one connected component that is a smooth plane quintic curve.

<B>Original post.</B>

<B>Moduli spaces of slope $0$ semistable sheaves with many isolated points</B>. Now consider the special case that $\mathcal{L} \cong \omega_{X/k}$.  In this case, the hypothesis is that $H^2(X,\mathcal{O}_X)$ equals $0$.  This is true if $X$ is a Calabi-Yau threefold, for instance, and if $C$ is a smooth elliptic curve.  In this case, since the determinant of $\mathcal{E}$ equals the trivial invertible sheaf, the defining short exact sequence proves that $\mathcal{E}$ is semistable.  Moreover, since $H^1(X,\mathcal{O}_X)$ also vanishes, also $H^1(X,\mathcal{E})$ vanishes.  

Thus, for every infinitesimal deformation of $\mathcal{E}$, there is a corresponding infinitesimal deformation of $s$, and thus an infinitesimal deformation of $C$, the zero scheme of $s$.  If $C$ is infinitesimally rigid, then it follows that $[\mathcal{E}]$ gives an isolated point of the moduli space $M$ of semistable sheaves with numerical invariants equal to those of $\mathcal{E}$.  Moreover, those numerical invariants depend only on the curve class of $C$ in $X$ and the genus of $C$, namely $1$.  Thus, if there is more than one infinitesimally rigid elliptic curve in $X$ having the specified curve class, then $M$ has more than one isolated point.

By the enumerative computation of Sheldon Katz mentioned above, a generic quintic threefold in $\mathbb{P}^4$ contains precisely $609250$ elliptic curves $C$ that are smooth plane cubics as curves in $\mathbb{P}^4$.  Thus, the corresponding moduli space $M$ of semistable sheaves of rank $2$ and slope $0$ has (at least) $609250$ isolated points.

<B>A moduli space of slope $0$ semistable sheaves with a connected  component of positive dimension</B> Next consider the special case of a smooth cubic threefold $Y$ in $\mathbb{P}^4$.  Consider the codimension $2$ closed subschemes of $Y$ that are lines $R$ contained in $Y$.  In this case, the moduli space of locally free $\mathcal{O}_Y$-modules of rank $2$ obtained from the Hartshorne-Serre correspondence has been thoroughly analyzed in the following article.

MR1795549 (2001j:14055) <br>
Markushevich, D; Tikhomirov, A. S. <br>
The Abel-Jacobi map of a moduli component of vector bundles on the cubic threefold. <br> 
J. Algebraic Geom. 10 (2001), no. 1, 37–62. <br>
https://arxiv.org/abs/math/9809140

The curves $R$ are genus $0$ curves with $\omega_{R/k} \cong \mathcal{O}_{\mathbb{P}^4}(-2)|_R$.  Thus, for the invertible $\mathcal{O}_Y$-module $\mathcal{L} \cong \mathcal{O}_{\mathbb{P}^4}(-2)|_Y$, the restriction of $\mathcal{L}$ to $R$ equals $\omega_{R/k}$.  Since $Y$ is a complete intersection $H^r(Y,\mathcal{O}(d))$ vanishes for every integer $d$ and for every integer $r$ that satisfies $0<r<\text{dim}(Y)$.  In particular, $H^{n-2}(Y,\mathcal{L})$ and $H^1(Y,\mathcal{L}^\vee\otimes_{\mathcal{O}_Y}\omega_{Y/k})$ both vanish.  Thus, by the Hartshorne-Serre correspondence, for every line $R$ in $Y$, there exists a unique pair $(\mathcal{F},t)$ of a locally free $\mathcal{O}_Y$-module $\mathcal{F}$ of rank $2$ and a global section $t$ whose zero scheme equals $R$.  Also, as in the case of elliptic curves on a Calabi-Yau threefold, $\mathcal{L}$ equals $\omega_{Y/k}$.  Thus, the locally free $\mathcal{O}_Y$-module $\mathcal{F}$ is semistable of rank $2$ and slope $0$, and $t$ is the unique global section up to scaling.  Thus, the component of the moduli space $N$ of rank $2$, slope $0$, semistable sheaves that parameterizes $\mathcal{F}$ (among others) is isomorphic to the Hilbert scheme of lines $R$ in $Y$.  This Hilbert scheme is a smooth surface of general type (cf. Clemens-Griffiths).  

<B>Moduli spaces of slope $0$ semistable sheaves with many components of positive dimension</B>. Finally, consider the product $k$-scheme $X\times Y$, where $X$ is a general qunitic threefold and $Y$ is a smooth cubic threefold.  The Picard group of $X\times Y$ is $\text{Pic}(X)\oplus \text{Pic}(Y)$, with ample cone equal to the product of the ample cones.  For any choice of ample divisor, the locally free sheaf $\mathcal{G}:=\text{pr}_X^*\mathcal{E}\oplus \text{pr}_Y^*\mathcal{F}$ is semistable of rank $4$.  By K\"{u}nneth computations, again $H^1(X\times Y,\mathcal{G})$ is zero.  Thus, the global section $s\oplus t$ deforms together with any infinitesimal deformation of $\mathcal{G}$.  The zero scheme of $s\oplus t$ is the codimension $4$ closed subscheme $C\times R$ in $X\times Y$.  Thus, the component of the moduli space of semistable sheaves on $X\times Y$ of rank $4$ and slope $0$ that contains $\mathcal{G}$ is isomorphic to a component of $M\times N$.   Since $M$ contains $609250$ isolated points $[\mathcal{E}]$ as above, and since $N$ is a smooth, projective, connected surface, it follows that this moduli space has at least $609250$ connected components that are each isomorphic to $N$, a smooth, projective, connected surface.