Is $DSPACE(f) \subseteq DTIME(f)$ always?
We don't know that $\mathrm{DSPACE}[f(n)]\subseteq\mathrm{DTIME}[f(n)]$ is ever true and we suspect that it's never true.

A Turing machine that uses space $f(n)$ can run for $2^{O(f(n))}$ steps without getting stuck in an infinite loop, so it should be able to do much more than a Turing machine that can only run for $f(n)$ steps.  As you've observed, if $\mathrm{DSPACE}[f(n)]\subseteq\mathrm{DTIME}[f(n)]$, then $\mathrm{DSPACE}[f(n)]=\mathrm{DTIME}[f(n)]$ and we have no reason to think that would ever be true for reasonable functions.