Why no fuamus?
There is an enormous difference in the frequencies of different forms of any word.
It is instructive to compare the more common versions first:
The form sit gives 6985 hits, while simus gives 232.
If we assume that the ratio of frequencies of the personal forms is the same for both variants (which may or may not be valid but is a reasonable ...
It's not that esse takes the accusative—it's that cupiō takes the accusative, and esse links two things in the same case. In other words, regem is accusative because mē is accusative, and mē is accusative as the object of cupiunt.
This is a fairly common construction in Latin, called the "accusative with infinitive" (or accusativus cum infinitivō ...
Cupere is a special kind of verb. You can use it to talk about something the subject of the sentence wishes to do himself. In that case you use an infinitive as the object and predicate nouns or adjectives are in the nominative:
Cupit rex esse.
He wants to be king.
Normal objects are in the accusative as usual, though:
Cupio placentam edere.
I want to eat ...
I think that the four examples from Ovid given by blagae are not quite relevant to the question raised by the OP: all of them can be argued to show a clearly adjectival behavior and are not infrequent at all in Classical Latin. It is not correct to say, as blagae does, that these examples "occur sporadically". One can apply the typical tests to ...