What characterizations exist for the grammars that can express subsets of the context-free languages?
Every context free language over a one letter alphabet (or equivalently every langauge recognized by unary PDAs, unary DPDAs or 1 counter machines) is regular.

See: S. Ginsburg, H. Rice: "Two families of languages related to ALGOL", Journal of the
ACM, 9: 350–371, 1962.

For what regards PDAs with bounded stack reversals they are equivalent to bounded context-free languages. In general a language $L$ over alphabet $\Sigma$ (with $|\Sigma|\geq 2$) is bounded if and only if there exists non empty words $w_1,w_2,...,w_n$ such that $L\ \subseteq w_1^*w_2^*...w_n^*$; but not all bounded languages are context-free (for a characterization of bounded CFG languages by finite union of stratified linear sets see Ginsburg's theorem). I don't know much about them, but they have been studied intensively, see for example: Oscar H. Ibarra, Bala Ravikumar , On Bounded Languages and Reversal-Bounded Automata (citing and cited articles).

EDIT: another result is that linear LR(1) languages are equivalent to the class of languages accepted by deterministic one-turn PDA (M. Holzer, K. J. Lange, On the Complexities of Linear LL(1) and LR(1) Grammars)