(1) is proved in Kashiwara-Schapira [KS], Theorem 8.6.5.
For (2), [KS], Proposition 8.6.11 prove that $\mathcal C$ is closed under kernels, cokernels, and extensions in $Ind(\mathcal C)$.
I believe the following is true, but I was not able to find any reference, unfortunately.
The condition is certainly often satisfied in practice.
Claim: We have that $\mathcal C$ is a Serre subcategory of $Ind(\mathcal C)$ if and only if every filtered colimit of subobjects of a fixed object $X \in \mathcal C$ exists in $\mathcal C$.
First, suppose the condition holds.
By above, it suffices to show that $\mathcal C$ is closed under subobjects in $Ind(\mathcal C)$.
Suppose we have a subobject $0 \to "\varinjlim_I\!\!" X_i \to X$.
The map is $"\varinjlim_I\!\!" (X_i \to X)$, and it's a general fact that we can compute finite (co)limits ``pointwise'',
in particular the image of this morphism is given by $"\varinjlim_I\!\!" X_i'$, where $X_i'$ is the image of $X_i$ in $X$ (see e.g. [KS], Lemma 8.6.4).
By assumption, $\varinjlim_I X_i'$ exists in $\mathcal C$, and finally
$$"\varinjlim_I\!\!" X_i \cong "\varinjlim_I\!\!" X_i' \cong \varinjlim_I X_i' \in \mathcal C,$$
as desired. In more detail, the second isomorphism holds because for any $Y \in Ind(\mathcal C)$ we have
$$Hom_{Ind(\mathcal C)}("\varinjlim_I\!\!" X_i',Y) = \varprojlim_I Hom_{Ind(\mathcal C)}(X_i',Y) = \varprojlim_I Hom_{\mathcal C}(X_i',Y) = Hom_{\mathcal C}(\varinjlim_I X_i',Y) = Hom_{Ind(\mathcal C)}(\varinjlim_I X_i',Y).$$
The converse follows easily, using the last displayed equation (the first two equalities, and restrict to $Y \in \mathcal C$).