Colimit and push forward of quasi-coherent sheaves in the analytic setting

I'm currently working on my master thesis and have some unresolved questions about quasi-coherent sheaves.
Since I'm new to algebraic geometry, they might be rather trivial.

I'm working in the analytic setting, i.e $(X,\mathcal{O}_X)$ will denote a complex manifold and its structure sheaf. I will use the definition from the stacks project, in which a sheaf $\mathcal{F}$ of $\mathcal{O}_X$ modules is quasi-coherent iff for every $x\in X$ there is an open neighbourhood $U$ of $x$, s.t. $\mathcal{F}|_U$ is isomorphic to the cokernel of a map $\bigoplus_{i \in I} \mathcal{O}_U \to \bigoplus_{j \in J} \mathcal{O}_U$.

1) Is the category of quasi-coherent sheaves on $X$ closed under colimits? Explicitly, I have a collection of finite rank vector bundles $\mathcal{W}^i\to X$ with embeddings $\mathcal{W}^i \hookrightarrow \mathcal{W}^j$ for all $i\leq j$ and would like to know, whether $\lim_{\rightarrow} \mathcal{W}^i$ is a quasi-coherent sheaf.

2) Let $f:X\to Y$ be a holomorphic map, $X$,$Y$ complex manifolds and $\mathcal{F}$ a quasi-coherent sheaf. Under which conditions on $X,Y,f$ is the push-forward sheaf $f_*\mathcal{F}$ a quasi-coherent sheaf. Explicitly, my $f$ is the projection $p:X\times Y \to X$, where $Y$ is a compact complex manifold and $X$ is any complex manifold and $\mathcal{F}$ is a direct limit of finite rank vector bundles (and if question 1 is true, a quasi-coherent sheaf).

I would be happy about any answers, partial answers and/or references, especially to papers/books which deal with quasi-coherence in the analytic setting.

I'm new to StackExchange, so feel free to correct my style/format, etc...

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–  Adeel Jul 9 at 14:04
@Adeel: Thanks for your answer. However, I am working in the analytic setting, i.e. $(X,\mathcal{O}_X)$ is a complex manifold and $\textit{not}$ a scheme. So my question is basically, whether there are similar results for this case. –  David Jul 9 at 14:23