Is there a Minesweeper solver?

Yes - there is one on this site. You enter the numbers you can see, and it works out which covered cells are guaranteed mines and which are guaranteed safe.

How a solver reaches its answer

Each visible number is a constraint: the covered cells touching it hold exactly that many mines. A solver enumerates every mine arrangement consistent with all of those constraints at once. If a cell holds a mine in every surviving arrangement it is provably a mine; if it is empty in every one it is provably safe. Anything else is genuinely undetermined.

That last case matters. A good solver tells you when a position cannot be resolved rather than picking the most likely cell and calling it an answer - if two cells are indistinguishable, no amount of computation separates them.

Is using one cheating

Not on a finished board or a position you are studying, which is the best way to use it: work the position out yourself, then check whether you were right. During ranked play it is a different matter - every ranked game here records its full move sequence, so solver-assisted play is detectable.

The solver you never see

No-Guessing mode runs the same kind of analysis before you get the board. Every board is generated and tested for solvability by pure logic; any board that would force a guess is discarded and replaced.