Are there established definitions of pseudomonads and pseudoalgebras on a (strict) 2-category? By pseudo, I mean that unitality and associativity hold up to some coherent isomorphism.
If so, are "free algebras" still a thing, i.e. does every object in the form $T X$ admit a pseudoalgebra structure given by the multiplication of the monad?
A reference would be welcome.
Just as one may define monads in a bicategory (equivalently 2-category), one may define pseudomonads in a tricategory (equivalently Gray category). The first comprehensive treatment was given in Lack's A Coherent Approach to Pseudomonads , though the definition itself goes back a little further, to Marmolejo, and Street. This theory was further developed in Gambino–Lobbia's On the formal theory of pseudomonads and pseudodistributive laws.
The objects of a Gray-category may be thought to correspond to bicategories, the 1-cells to pseudofunctors, and the 2-cells to natural transformations. However, because we may be working in a non-Bicat-like Gray-category, the objects may not themselves have an internal notion of "object", so this question doesn't make sense in this general context. However, given enough colimits, it is possible to form Kleisli objects corresponding to the bicategories of free pseudoalgebras, just as in the 1-dimensional setting (that is, for monads in 2-categories).
If you are interested instead in pseudomonads on a 2-category (the distinction between the two propositions is easy to get overlook, but important in this case), then pseudomonads act very similarly to monads, and free pseudoalgebras can be described similarly to the 1-dimensional case. The definition of a pseudoalgebra, along with its coherence conditions, is described explicitly in Section 4.2 of Lack's A Coherent Approach to Pseudomonads. I'm not sure where the definition of a free pseudoalgebra is spelt out explicitly, but it may be derived directly from the definition of a biadjunction: in any case, the data should be straightforward to work out from the definition of a free algebra and a pseudoalgebra.