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 $2$-categorical initial object. Otherwise, $\{1\}$ would be $2$-categorical initial, i.e. the category of monoidal functors $\{1\} \to \mathcal{C}$ would be 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 $1 \to M$.
Essentially, the notion of isomorphism of categories is often too strict.
A more interesting example: In the $2$-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 $2$-categorical initial object. If $R$ is a commutative ring, then $(\mathbf{Mod}_R,\otimes)$ is bicategorical (and not $2$-categorical) initial in the $2$-category of cocomplete symmetric monoidal $R$-linear categories.
If $X,Y$ are two quasi-compact quasi-separated $R$-schemes, then $\mathbf{Qcoh}(X \times_R Y)$ is the bicategorical coproduct of $\mathbf{Qcoh}(X)$ and $\mathbf{Qcoh}(Y)$ in the $2$-category of cocomplete symmetric monoidal $R$-linear categories (this is the main result of Localizations of tensor categories and fiber products of schemes), but there will not be any $2$-categorical coproduct.