What is the simplest known NP-Complete problem for testing P=NP solutions?
Since all NP-complete problems are basically equivalent, it's hard to say which one is the easiest. SAT was one of the original problems and has important practical applications, so it is really well studied. Writing fast SAT-solvers seems like a reasonably interesting hobby to me and getting started is not terribly hard. Integer Linear Programming is similarly important and well studied.

The only objective way to discern NP-complete problems that I know of is their approximability. Some problems are hard to approximate, for others you can get almost arbitrarily good solutions in polynomial time.

Subset sum is both very simple to understand and "simple" to solve. Simple to solve means that it has an easy Pseudo-polynomial algorithm and is easy to approximate. However, I don't know about any research involving the problem.