Exponentiating a representation and Baker–Campbell–Hausdorff formula

Let $$G = \{\rho \in \mathrm{Aut}(\mathbb{C}[[t]]) \,|\, \rho(t) \in t + t^2 \mathbb{C}[[t]] \}$$ be a subgroup of continuos $$\mathbb{C}-$$automorphisms of $$\mathbb{C}[[t]]$$ and $$\mathfrak{g} = t^2 \mathbb{C}[[t]] \, \partial_t \subseteq \mathrm{Der}(\mathbb{C}[[t]])$$. Define $$\exp: \mathfrak{g} \to G$$ by the usual exponencial series $$\exp(v(t) \partial_t) = \sum_{n \geq 0} \frac{1}{n!} (v(t) \partial_t)^n$$ This map is well-defined bijection. Let $$r$$ be a locally nilpotent representation of $$\mathfrak{g}$$, i.e., $$r: \mathfrak{g} \to \mathfrak{gl}(V)$$ such that for all $$X \in \mathfrak{g}$$ and $$a \in V$$ we have $$r(X)^n a = 0$$, $$n\gg0$$. I want to define a representation $$R: G \to \mathrm{GL}(V)$$ exponentiation $$r$$, i.e., $$R(e^X) = e^{r(X)}$$. As $$\exp: \mathfrak{g} \to G$$ is a bijetion and $$r$$ is locally nilpotent, we can define such $$R$$.

Question 1:: I want to prove that $$R$$ is a group homomorphism.

Let $$w(t) \partial_t \in \mathfrak{g}$$ s.t. $$e^{w(t) \partial_t} = e^{u(t) \partial_t} e^{v(t) \partial_t}$$. It is sufficient to prove that $$w(t)$$ is in the Lie subalgebra generated by $$u(t)$$ and $$v(t)$$. I would like to prove it without BCH formula. I could find the coefficients of $$w(t)$$ in terms of $$u(t)$$ and $$v(t)$$, but it's not an easy formula.

Question 2:: Are those all representations of $$G$$ ?

If $$V$$ is finite dimensional (over $$\mathbb{C}$$) I guess yes. Take $$r(X) = \left.\frac{d}{dt}\right|_{t=0} R(e^{tX}) \in \mathfrak{gl}(V)$$ (if $$V$$ is finite dimensional this derivative make sense), it is easy to see that $$R(e^{X}) = e^{r(X)}$$.

If $$V$$ is the limit of it's finite dimensional subrepresentation I think it will work too.