There is a large combinatorial literature on this, with many extensions and applications (including, notably, the MacMahon Master theorem). I am giving below a few pointers so you can start - the real literature is very large.
2) $q$- and non-commutative extensions. There are several $q$-analogues due to Andrews (see here, Garsia (see here for ref and bijective proof), and Gessel (see here for both $q$- and non-commutative generalization). Finally, see this (more general) non-commutative extension using quasideterminants.