The answer to your second question is also no: only local properties can be characterised in the internal language of a topos, and it is possible to have locally constant (pre)sheaves that are not constant.

A simple, and well-known, example runs as follows: let $\mathbb C$ be the poset with four elements $n$, $e$, $s$, and $w$, satisfying $w<n$, $w<s$, $e<n$, and $e<s$.
Now let $P$ be the presheaf on $C$ which maps: each object to $\{0,1\}$; all but one arrow (say, the one from $w$ to $n$) to the identity; that one other arrow to the non-identity automorphism of $\{0,1\}$.
This presheaf is not isomorphic to a coproduct of copies of 1, and therefore not constant.
(If you draw it right, $P$ looks like the non-trivial double cover of the circle.
In fact, this is not just a metaphor: $\hat{\mathbb C}$ can be construed as the topos of sheaves on a certain non-Hausdorff quotient of the circle.)

Now let $N$ and $S$ denote Yoneda of $n$ and $s$, respectively; then $N+S$ has global support. The slice topos $\hat{\mathbb C}/(N+S)$ is equivalent to $\hat{\mathbb D}$, where $\mathbb D$ is the poset having elements $n,e_N,e_S,s,w_S,w_N$, satisfying $w_N<n$, $w_S<s$, $e_N<n$, and $e_S<s$. Moreover $(S+N)^*:\hat{\mathbb C}\to\hat{\mathbb C}/(N+S)\simeq\hat{\mathbb D}$ maps $P$ to the presheaf (let's call it $Q$) which maps: every element of $\mathbb D$ to $\{0,1\}$; all but one arrow ($w_N \to n$) to the identity; that one arrow to the non-identity automorphism of $\{0,1\}$. Unlike $P$, $Q$ actually is isomorphic to a coproduct of copies of $1$, and therefore constant.
(Continuing the parenthetic remark at the end of the previous paragraph: we have trivialised the non-trivial double cover by breaking the circle into two semicircles as usual.)

The larger point here is that $(S+N)^*$, being the inverse image functor of a surjective local homeomorphism, is both logical and faithful; logical functors preserve interpretations of formulas, and faithful ones reflect validity. So if $\Phi$ is a formula which is true for all constant sheaves, its validity for $Q$ entails its validity for $P$.
In fact, this applies for infinitary formulae as well, since $(S+N)^*$, having both a left and a right adjoint, preserves arbitrary limits and colimits.

Now one may wonder: can locally constant sheaves be characterised by an internal formula? I think that the answer is yes, but the man to ask is Thomas Streicher. Both he and Richard Squire were (independently) interested in this question in the late '90s and (I think) reached answers in the early '00s. I am relatively sure that Richard did not publish his results, but perhaps Thomas did? This was about the time I was losing interest in topos theory, though I have picked it up again recently.

What I can remember is this:

- locally constant presheaves are the same as presheaves valued in {sets-with-bijections};

[1.5 decidable presheaves are the same as presheaves valued in {sets-with-injections};]

- (for presheaves) it is therefore enough to characterise "transition-surjective" presheaves---i.e., presheaves valued in {sets-with-surjections};

[2.5 Kuratowski-finite presheaves are precisely those which are both transition-surjective and finitely-valued;]

- the problem of characterising transition-surjective presheaves is therefore equivalent to that of characterising finitely-valued presheaves, which is an interesting question in its own right.

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