I would argue that the "correct" version of (co)limits is the "pseudo" or "bi" or "weak" or "strong" or "homotopy" version. (As far as I can tell, all of these words mean the same thing.) The bicategories of cocomplete or of locally presentable categories are closed under limits: indeed, you can show by hand that the limit just as categories of cocomplete categories along cocontinuous functors is itself cocomplete, and Bird's thesis verifies that if all constituents are locally presentable, then so is the limit. Lurie has shown the same statement in the $\infty$-categorical world.

2-rings, in either Brandenburg's sense or ours, are therefore the symmetric monoidal objects in a complete bicategory. Just as in the 1-categorical case, I believe it is straightforward (if tedious) to show that the limit of underlying categories carries a canonical symmetric monoidal structure, making it the limit of 2-rings. So this handles the "limit" version of your question.

The colimit version I believe also works, although you cannot just take the colimit of underlying categories. I haven't thought as much about the bicategory of all cocomplete categories, but the bicategory of locally presentable categories is known, again by Bird's thesis, to contain all colimits. (They are computed by computing instead the limit along right adjoints to the cocontinuous functors you have in mind.) Then you can present a colimit of 2-rings (in our sense) much as you would present a colimit of commutative algebras: any time you see a coproduct, for example, write down a free commutative algebra; any time you see a quotient, do it symmetric monoidally; etc. for the other types of colimits in the bicategorical world. I haven't done the exercise fully myself, but again at least in the locally presentable case I'm sure you can do it (in terms of sketches if necessary).

I don't recall off the top of my head whether colimits of arbitrary cocomplete categories exist. I could imagine that you would run into size problems, but I don't know.

I should say, there is also a fair amount of work on the bicategorical version of monads, and (co)limits, in all senses (including lax and oplax), of their algebras. I don't recall the references, but my memory is that there are no surprises from the 1-categorical case.