It is hard to say what the non-physical is. However, it can be detected. For example, if we could, say, solve the halting problem, then whatever part of us is doing the solving cannot be physical, since everything physical can be computed by a Turing machine.
So, in general the non-physical is defined negatively. The physical is defined by physical limitations, i.e. the laws of physics and computation, and thus anything that surpasses those limitations is non-physical.
There is no algorithm for checking whether a system can solve the halting problem or not, the reduction to the halting problem is quite simple. But that's besides the point.
If you found something that could solve the halting problem and you're lucky enough to be able to prove it for this special case, that would just mean that the assumption that everything physical can be computed by a Turing machine is wrong. It works the same way with anything else that we think of as "impossible". If you observed an apple falling towards the sky that doesn't mean that the apple is non-physical, it just points to a serious flaw in our understanding of gravity. Experiments trump all physical theories.
Technically, we already recognize the world is non-physical. Materialists defined materialism to things bumping into each other. Now we know there are things like field effects, attraction, and action at a distance. So there is a non-bumping substrate that the bumping things exist within. Thus, strict physicalism is already known to be false.
No it doesn't. It just implies that everything that causally interacts with the observable reality is physical. It leaves open the door for an immortal soul, it's just that the existence or nonexistence of the soul has absolutely no effect on the observable universe.
The term physicalism was introduced a few decades after we figured out Maxwell's equations. If the philosophers defined it to mean things bumping into other things they were frankly poorly informed about the world we live in.
So, in general the non-physical is defined negatively. The physical is defined by physical limitations, i.e. the laws of physics and computation, and thus anything that surpasses those limitations is non-physical.