Gee, that's nice.
Well, then a starting note on my end: if anyone of us suggests certain components, it's your job to find them, so don't start complaining about 'but it's difficult to get...' Consecutively, as to the story about the logic level mosfets: those are what you need. So stop rabbling about using IRF520/540. Those are unfit for most of your purposes, hopelessly outdated and not worth buying at this point.
Usually, unless you have circuits where one part is lifted with respect to another, or one part has no ground reference that is used as a circuit ground. Think e.g. a power supply that plugs into a wall outlet.
Concerning the methods of grounding - there's several topologies. Ground planes, star grounds, bus grounds. Each have their pros and cons and applications.
No. The module itself should include one, but that's not what you ask, I assume.
There can be difference in internal topology that make one more suited to a certain application than another. E.g. the output of a comparator may be open collector/open drain, making it relatively 'agnostic' to what voltage levels it interfaces with on the output, but not capable of sourcing current. But in practice, I don't bother with sourcing dedicated comparators and just use general purpose opamps.
Then you probably didn't look closely enough. I haven't come across opamp (or other component...) datasheets that don't specify permissible input voltages. Take another look, it's there. So no, you can't just assume that something works the way you expect because it isn't specified - which, like I said, it really is in this case. An opamp datasheet would be virtually useless if it doesn't specify the limits to input voltages. Even rail-to-rail opamps specify margins (offsets).