Q1. Yes, every object that is equivalent to a bicategorical limit is again a bicategorical limit. This is easy to deduce from the definition. It can also be deduced from the corresponding fact about representable pseudofunctors.

Q2. The 2-category of monoidal categories has a bicategorical initial object $\{1\}$, but no $2$-initial object. Otherwise, $\{1\}$ would be $2$-initial, i.e. the category of monoidal functors $\{1\} \to \mathcal{C}$ would be the isomorphic to the terminal category $\star$ for all $\mathcal{C}$. But this is not the case:

It is well-known that the lax monoidal functors $\{1\} \to \mathcal{C}$ are monoid objects $(M,m,e)$ in $\mathcal{C}$. The functor is strong monoidal iff $e : 1 \to M$ and $m : M \otimes M \to M$ are isomorphisms. It follows that the category of strong monoidal functors is isomorphic to the category of isomorphisms $1 \to M$. This is clearly equivalent to $\star$, but not isomorphic to $\star$, since there are many isomorphisms $1 \to M$.

A more interesting example: In the category of cocomplete symmetric monoidal categories (this includes the condition that $\otimes$ is cocontinuous in each variable), $(\mathbf{Set},\times)$ is bicategorical initial, but there is no $2$-initial object.