We have $X = {\rm Spec}({\rm Sym}(T_{\mathbb{P}^2}))$, and the ideal of the zero section is cut out by the first grading i.e. $T_{\mathbb{P}^2}$. Now, the tangent sheaf on $\mathbb{P}^2$ is globally generated. Thus, if $E$ is supported set theoretically on the zero section, then $\Gamma(\mathbb{P}^2, T_{\mathbb{P}^2})$ acts on it by nilpotent endomorphisms, and if ${\rm Hom}(E, E) = \mathbb{C}$, then all of these have to be zero, i.e. $E$ is (scheme-theoretically) supported on the zero section, $E = i_* i^* E$.

Sure, there is a whole area of differential Galois theory, in which the Galois groups are linear algebraic groups. The category of finite extensions of a field is replaced with that of finite differential modules. In characteristic zero, this is a Tannakian category and thus is is the category of representations of a proalgebraic group.

@Z.M $f$ is a proper morphism between varieties over a field, and hence is of finite presentation.

The underlying vector bundle of a $\mathbb{C}$-local system is always algebraic, by Deligne's Riemann-Hilbert correspondence (LNM 163).

Correction: it should of course be $\Delta - (E\times 0)$. This divisor has self-intersection $-2$, and hence $\chi(L) = 1$ by Riemann-Roch. It follows that $R^i f_* (E)$ cannot be locally constant, because if so, they would all be zero, and then $\chi(L)=0$ by the Leray spectral sequence.

No, it does not. Let $Y = E$ be an elliptic curve and let $X = E\times E$ with $f\colon E\times E\to E$ the first projection. Consider the line bundle $L = \mathcal{O}_X(\Delta+(E\times 0))$ where $\Delta$ is the diagonal. Its restriction to the fiber above a point $P$ is the line bundle $\mathcal{O}_E(P - 0)$ of degree zero. It has a section if and only if it is trivial, i.e. for $P = 0$. So the dimension is not constant for $i=0$ or $i=1$.

Yes, because it is the analytification of the vector bundle obtained from $\mathbb{L}\otimes\mathcal{O}_{X_{\rm et}}$ by descent (vector bundles on $X_{\rm et}$ and on $X$ are the same). Indeed, there is a natural map between the two, and to check it is an isomorphism, we may do so etale locally on $X$, and therefore reduce to the case $\mathbb{L}$ being constant, in which case it's obvious.

No, this is not true, there are plenty examples in the literature, starting with Serre's 1961 paper.

I don't understand the question – in your situation, the torus is $T={\rm Spec}(k[\mathbb{Z}^n])\simeq \mathbb{G}^n_m$ which is split.

Differentials and cotangent bundles work roughly in the same way as for varieties or schemes.

No, the maximal ideal of $k[[x,y]]$ is torsion-free but not flat.