Dual display setup question

Yes, of course - but only if you have "Displays have separate Spaces" checked.

FWIW, ever since this option exists (I think it may have been since Mavericks 10.9), I always have it checked.

Right now, I have Safari in full screen on one display, and Safari, and TextEdit, also in full screen, on the other. As a matter of fact, you can have several instances of the same or different applications in full screen. Each full screen application automatically gets its own Space.

Other than putting some application in Full Screen on the second display, no.

When "Displays have separate Spaces" is checked, each display has its own menu bar - it is a separate Space, after all: just like when you have several Spaces on the same display, each has a menu bar too.

That would be very confusing… you switch to another Space and suddenly there is no menu bar…

But I forgot something that may or may not suit your wishes: you can of course simply decide to hide the menu bar, so that it only pops up when you move the cursor to the top of your screen. This is all or nothing, though, you can’t set it separately for different Spaces.

Sure, just not use more than one.

However, the second display issue remains.