We consider the deformation theory of a generalized elliptic curve $(C_0,+)$ over a field $k$. Let $D$ be the deformation functor.

And now we only consider the case that $C_0$ is irreducible as in D-R (Deligne-Rapoport) Chapter III, 1.3. Then $D$ is equivalent to the deformation functor $D'$ of the couple $(C_0,e)$, with $e$ a marked smooth point ($e$ corresponds to the identity of the law $+$). In D-R, it says that $\mathrm{Ext}^1(\Omega_{C_0}(e),\mathcal{O}_{C_0})$ is the space of first order deformations and the obstruction for infinitesimal lifting lives in $\mathrm{Ext}^2(\Omega_{C_0}(e),\mathcal{O}_{C_0})$.

Could someone explain why $\Omega_{C_0}(e)$ plays such a role? Or maybe you could point out the reference to me.

Another related question is: we know from D-R, $D$ is prorepresented by $(R,\xi)$ where $R$ is complete noetherian $\Lambda$-algebra with residue field $k$ and $\xi=\varprojlim \xi_n$ with $\xi_n\in D(R/m_R^{n+1})$, $\xi$ gives rise to a formal scheme $\mathcal{C}=\varinjlim C_n$ over $\mathrm{Spf} R$ with $C_n$ corresponding to $\xi_n$. Since $C_0$ is projective, Grothendieck's formal existence theorem garantees a unique projective curve $C$ over $\mathrm{Spec} R$ whose formal completion is $\mathcal{C}$.

My question is why $C$ admits a structure of generalised elliptic curve? Or equivalently why the 'group' law algebraizes?

I should read D-R carefully. Question 2 follows from D-R Chapter II Prop. 1.5, Prop. 2.7. However, I have a similar question for toroidal models of abelian varieties, see Semiabelian actions appearing in the toroidal campactification of a degenearting abelian varieties

Remark: I say "similar" because both are about lifting group action from formal world to algebraic world.