For Q1: this is dealt with in a context more general than the classical one by Adamek, Borceux, Lack and Rosicky in their paper "A classification of accessible categories". They replace finite or $\kappa$-small limits with an arbitrary class of limits $\mathbb{D}$, and consider a condition which they call soundness, one of whose consequences is a decomposition of every $\mathbb{D}$-flat weight as a suitably 
"$\mathbb{D}$-filtered" colimit of representables.

This is all in the unenriched context, which is not what you want, but the point is that they make axiomatic assumptions which are more or less exactly what is needed to force the answer to your question 1 to be true. Make of that what you will, but it at least implies that it's not automatic, and will probably require a bespoke argument in each situation.

For Q2: No. I guess the classical reference is Kelly's "Structures defined by finite limits in the enriched context." If $\mathcal V$ is a symmetric monoidal closed category which is locally $\kappa$-presentable as a closed category (i.e., it is locally $\kappa$-presentable and the $\kappa$-presentable objects are closed under the monoidal structure), then there is a good notion of locally $\lambda$-presentable $\mathcal V$-category: they are precisely the cocomplete $\mathcal V$-categories, whose underlying ordinary categories are locally $\lambda$-presentable, and whose $\lambda$-presentable objects are closed under tensors (=copowers) with $\lambda$-presentable objects of $\mathcal V$. Without this last condition, there is a gap through which to thread a negative answer to your question.