I wanted to flesh out Martin's nice concrete examples to Question 2 a bit with a more general perspective. As with the 2-category of monoidal categories and strong or lax monoidal functors, basically every 2-category of structured categories and pseudo, lax or oplax morphisms will be bicategorically complete and cocomplete but not 2-categorically so. This includes 2-categories like the 2-category of categories admitting $J$-shaped limits or colimits and functors preserving the same as well as various relatives of monoidal categories, and quite generally, the 2-categories of algebras and pseudo-morphisms for any 2-monad $T$ on 2-categories such as those of categories or of multicategories.

These 2-categories, as well as strict 2-initial or 2-terminal objects, lack strict (co)equalizers, pullbacks, and pushouts. For a simple example, let $\star$ be the terminal category and $I$ the category freely generated by an isomorphism. These are both trivial examples of categories with finite products, and both functors $\star\rightrightarrows I$ preserve these products (for instance, because they are equivalences of categories.) But not only does this diagram have no equalizer, there is no object whatsoever in the 2-category of categories with finite products that strictly equalizes these two morphisms! This behavior is quite typical; note that however $\star$ is the equalizer of this diagram in the bicategorical sense.

A good keyword for learning more on this topic is "PIE": a 2-category admitting the specific set Products, Inserters, and Equifiers of weighted 2-limits is always bicategorically complete.

As for Question 3, I commented on the main post that I'm not quite sure that a strict weighted bicategorical limit is a well-defined notion. However certainly it's true that pseudo- lax- and oplax-limits, in the bicategorical sense, can also be given in terms of weighted limits (themselves defined in the pseudo sense.) I think I would approach this by observing that not only does every bicategory have a strictification, i.e. an equivalent 2-category, but it has a continuously equivalent 2-category, namely its image under the bicategorical Yoneda embedding, which is a complete 2-category for which the Yoneda embedding preserves whatever limits exist. So you can convert, say, a bicategorical pseudolimit to a weighted one by converting the corresponding 2-categorical pseudolimit to a weighted one under the Yoneda embedding.

I wanted to flesh out Martin's nice concrete examples to Question 2 a bit with a more general perspective. As with the 2-category of monoidal categories and strong or lax monoidal functors, basically every 2-category of structured categories and pseudo morphisms will be bicategorically complete and cocomplete but not 2-categorically so. This includes 2-categories like the 2-category of categories admitting $J$-shaped limits or colimits and functors preserving the same as well as various relatives of monoidal categories, and quite generally, the 2-categories of algebras and pseudo-morphisms for any 2-monad $T$ on 2-categories such as those of categories or of multicategories.

These 2-categories, as well as strict 2-initial or 2-terminal objects, lack strict (co)equalizers, pullbacks, and pushouts. For a simple example, let $\star$ be the terminal category and $I$ the category freely generated by an isomorphism. These are both trivial examples of categories with finite products, and both functors $\star\rightrightarrows I$ preserve these products (for instance, because they are equivalences of categories.) But not only does this diagram have no equalizer, there is no object whatsoever in the 2-category of categories with finite products that strictly equalizes these two morphisms! This behavior is quite typical; note that however $\star$ is the equalizer of this diagram in the bicategorical sense.

A good keyword for learning more on this topic is "PIE": a 2-category admitting the specific set Products, Inserters, and Equifiers of weighted 2-limits is always bicategorically complete.

As for Question 3, I commented on the main post that I'm not quite sure that a strict weighted bicategorical limit is a well-defined notion. However certainly it's true that pseudo- lax- and oplax-limits, in the bicategorical sense, can also be given in terms of weighted limits (themselves defined in the pseudo sense.) I think I would approach this by observing that not only does every bicategory have a strictification, i.e. an equivalent 2-category, but it has a continuously equivalent 2-category, namely its image under the bicategorical Yoneda embedding, which is a complete 2-category for which the Yoneda embedding preserves whatever limits exist. So you can convert, say, a bicategorical pseudolimit to a weighted one by converting the corresponding 2-categorical pseudolimit to a weighted one under the Yoneda embedding.