Rank of a jet bundle of a vector bundle. Interpretation of the first jet bundle

Let me just complete Alex' answer to 3). First of all, the jet bundles have nothing to do with any structure group; they are associated to vector bundles, period. Then $J_1(E)$ fits into an exact sequence: $$0\rightarrow \Omega ^1_X\otimes E\rightarrow J_1(E)\,\buildrel {e}\over {\longrightarrow} \,E\rightarrow 0\ .$$ At each point $p\in X$, with maximal ideal $\mathfrak{m}_p\subset \mathcal{O}_{X,p}$, a 1-jet is just a function (in a neighborhood of $p$) modulo those vanishing at order $\geq 2$ at $p$, that is, modulo $\mathfrak{m}_p^2$. The homomorphism $e$ associates to such a function its value at $p$; associating to a function vanishing at $p$ its differential gives the isomorphism of $\mathrm{Ker}( e)$ with $\Omega ^1_X\otimes E$.

This exact sequence plays an important role: its extension class $\mathrm{at} \in \mathrm{Ext}^1_{\mathcal{O}_X}(E,\Omega ^1_X\otimes E)\cong H^1(X,\Omega ^1_X\otimes \mathcal{E}nd(E))$ is the famous Atiyah class. The vanishing of that class is a necessary and sufficient condition for the existence of a section, which is equivalent to the existence of a holomorphic connection on $E$. The Chern classes of $E$ in $\ \oplus\, H^p(X,\Omega ^p_X)$ can be constructed from $\ \mathrm{at}\ $ by applying invariant polynomials to $\mathcal{E}nd(E)$.

(1) Locally, jets of sections are just collections of $r=\operatorname{rank}E$ jets of functions, hence, the rank of $J_k(E)$ equals $r$ times the number of multiindices $I=(i_1,\ldots,i_n)$ with $|I|\le k$.

(2) It is certainly holomorphic.

(3) It seems to me that $J_1(E)=T^*X\otimes E$.

Please let me add something about the question of holomorphicity:

If $E$ is a holomorphic vector bundle, then there is a bundle of jets of holomorphic sections, which is holomorphic, as explained by @hm2020. The bundle of 1-jets of smooth sections of $E$, on the other side, does not have a holomorphic structure.

Example: let $E$ be a trivial bundle over $X$, so that the exact sequence described by @abx splits. Then the jet bundle of smooth sections of $E$ is $$(T^*_{\mathbb R}X\otimes_{\mathbb R} E) \oplus E = (\Omega^{1,0}(X)\otimes_{\mathbb C} E) \oplus (\Omega^{0,1}(X)\otimes_{\mathbb C} E) \oplus E~.$$ While $(\Omega^{1,0}(X)\otimes E)\oplus E$ has a holomorphic structure and identifies with the bundle of jets of holomorphic sections, the other summand $\Omega^{0,1}(X)$ does not have a preferred holomorphic structure.