Are there undecidable properties of non-turing-complete automata?
Undecidable problems about context free grammars, and hence, pushdown acceptors as well, which are restricted TMs from Wikipedia...


Given a CFG, does it generate the language of all strings over the alphabet of terminal symbols used in its rules?
Given two CFGs, do they generate the same language?
Given two CFGs, can the first generate all strings that the second can generate?


There are many others about CFGs/PDAs as well as CSGs/LBAs and many other "simpler than TM" models.