SAT Solving + Turing Machines
Yes, it is common for a SAT solver to combine several techniques, e.g., random restarts with smart backtracking or other tricks. More generally, nothing stops you from throwing everything and anything at a problem you want to solve. (The reason a book might talk very specifically about say "probabilistic algorithms" instead of "here's how you solve problem X by using a probabilistic algorithm in combination with this trick, that heuristic and this data structure" is that it wants to keep focus on a specific algorithmic idea and its analysis).

The key point here is that a SAT solver is intended to be used in practice, and this is why it makes sense to combine several algorithmic techniques, data structures, etc. to achieve practical efficiency. Still, in general, no matter what the solver does, it cannot be fast and correct on every possible instance (due to NP-completeness).

To understand why having a fast algorithm for an NP-complete problem allows you to solve other such problems, look up polynomial-time reductions, and read up on NP-completeness. Here, again, it is useful to notice that all of this is perfectly fine in theory, but in practice things might be different (it is not true that "polynomial time" necessarily means "efficient in practice").