If my primary monitor is to the right of my secondary monitor, do applications open on the primary monitor or the 'left' monitor? Also, do applications remember which monitor they were last opened on?

That depends on the DE.

On Unity it opens on the monitor you're focusing.

Sometimes it's a bit tricky. Let's say you're focusing on a Window on your main monitor but your mouse cursor is in the secondary monitor. If you hit the meta key, the menu will show on the secondary monitor. Now, if you open an application without using your mouse it'll open on your main monitor, where you had the focused window but if you click on the app it'll open on the secondary monitor because the click changes the focus.

Some stuff is not as polished as it should be but if we take in consideration that Unity has been abandoned for 4 years and before that it wasn't developed at all as all the effort was put into Unity8, I'd say it's a pretty damn good DE.

I hated it when it came out but now I can't live without it.

Two 4k monitors with Ubuntu Mate, works well.