$\mathsf{PSPACE}$ membership in exponentially many steps?
whether it is ok to have an algorithm that takes an exponential number of steps as long as, overall, the whole thing stays in polynomial space.


Yes, of course. Polynomial space means exactly what it says.

The only reason you should worry is if you think that your algorithm takes more time than $2^{n^c}$ for all&nbsp;$c$.  In that case, it must either use more than polynomial space or loop forever since, there are only $|\Sigma|^{n^k}$ configurations of a tape of length&nbsp;$n^c$ using alphabet&nbsp;$\Sigma$, only $n^k$ possible head positions and only a constant number of possible states. That means there are only $|Q|n^k|\Sigma|^{n^k} \leq 2^{n^c}$ (for large enough&nbsp;$c$) possible configurations for a polynomial space Turing machine so if it runs for more than that many steps, it must have seen the same configuration twice so it must be in an infinite loop.