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, but the category of monoidal functors $\{1\} \to \mathcal{C}$ is not isomorphic to $\star$, it is only equivalent to $\star$.