1. The creation of weighted limits by the forgetful functor from the $\mathscr V$-category of algebras for an enriched (relative) monad is proven in Proposition 2.5 of Arkor–McDermott's [Relative monadicity](https://arxiv.org/abs/2305.10405). I'm not aware of a reference for the creation of colimits.

2. The creation of bilimits by the forgetful pseudofunctor for a pseudomonad is proven in Theorem 6.3.1.6 of Osmond's [A categorical study of spectral dualities](https://hal.science/tel-03609605v4). (The creation of conical bicolimits that are preserved by the pseudomonad is also claimed in Lemma 2.7 of Osmond's [Codescent and bicolimits of pseudo-algebras](https://arxiv.org/abs/2204.06055) without proof.)