What complexity class is this set of grammars?
Every unrestricted grammar can be converted to one with the following types of rules:
$$
AB \to CD \\
A \to BC \\
A \to a \\
A \to \epsilon
$$
Wikipedia calls this Kuroda normal form for unrestricted grammars.

You can convert a rule of the form $AB \to CD$ to a pair of rules $AB \to X_{ABCD}$, $X_{ABCD} \to CD$. Following the OP's suggestion, we can convert $A \to \epsilon$ to the rule $A \to E$, and add rules $BE \to B$ and $EB \to B$ for any nonterminal $B$. This allows us to obtain a grammar in your form equivalent to the original one, up to not being able to generate $\epsilon$.