The method will vary from one system to another, as well as based on option settings.

Definite maybe, if it is a system of polynomials. You can do

NSolve[a]

to get the code for NSolve loaded. Then do

SystemNSolveDumpdebugLevel=12;

With this set, you might get a spew of messages when you feed NSolve your system. If you do, it means NSolve at least used the eigensystem decomposition method in some part of the process. If not, then it did "other things" (apologies for such technical jargon).

Note that the variable being set is strictly for internal debugging purposes. You are using it at your own risk. I don't think it will bite. But bear in mind I'm the owner of this particular pet and it has never interacted with outsiders in my presence, so I can't really predict the outcome.

Yes, it means some other method or methods were used. For polynomial systems almost certainly it would be the sparse polyhedral homotopy method.

NDSolve has considerable support for method options and suboptions. Also it has step and event monitors that can record information along the way.