Problems of similar complexity for different measures
The answer to the first question is that we don't know, because we don't know whether $\mathbf P=\mathbf L$, so it could be that all $\mathbf P$ problems use only logarithmic space.

For the second one, most reasonable problems need at least linear time to read the input, even for non-deterministic machines, so it does not make a lot of sense to say "polynomial non-deterministic lower bound", or you need to make your question more precise.

Finally, the last question is like the first one: it could be that $\mathbf{NL}=\mathbf{NP}$, in which case such a problem wouldn't exist, we don't know...