Edit distance in sublinear space
Just to get things going, rather than trying to close out this problem: there is an obvious nondeterministic algorithm using logarithmically many bits of space (search for a single path through the dynamic programming matrix) so by Savitch's theorem there is a deterministic algorithm with space $O(\log^2 n)$. Its time must be of the form $n^{O(\log n)}$, quasi-polynomial rather than exponential.

There are some space lower bounds for edit distance in http://arxiv.org/abs/1106.4412 but I don't think they match your version of the problem.