Is there a logspace version of the reduction used in Cook-Levin?
To address your "follow-up", no.  The ability to use logspace reductions instead of polynomial time reductions to prove NP-completeness doesn't prove that P is contained in&nbsp;L.  It just proves that you don't need the full power of polynomial time reductions to define NP-completeness.  The situation is similar to saying, "Hey, there are lots of polynomial-time algorithms for graph reachability but I just found out that the problem is in logspace.  Does that mean that P$\,=\,$L?" except that now we're talking about a whole class of computations (the reductions) rather than just one problem.

In fact, if you check out books on descriptive complexity,1 you'll see that we can use even weaker notions of reduction (reductions definable by first-order formulas) and still define a meaningful and, as far as we know, equivalent notion of NP-completeness.  This is reassuring: it tells us that NP-completeness is a robust concept that doesn't seem to depend exactly on how you define it.

I say "doesn't seem to depend" because, as far as I'm aware, it is not known that defining NP-completeness in terms of polynomial time, logspace and first-order reductions makes exactly the same class of problems NP-complete.  But there are no problems known to be NP-complete under one of these kinds of reductions but not under one of the others.  (Stand by for this paragraph to be edited after people comment!)



1 E.g., Immerman, Descriptive Complexity (Springer, 1999) or Libkin Elements of Finite Model Theory (Springer, 2004).