Yes, applications remember which monitor and even location on the monitor they were last at.
I'll probably give Budgie a go.
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.
To be fair, my Thinkpad from work is the first laptop that has issues with when (dis)connecting a screen. When disconnecting I need to turn on the laptop screen before disconnecting; for connecting I can usually log in without screen and use Super+P to switch modes and that fixes it.
But other than this first instance, I never heard of this being an issue and people here talk about it as if it's commonplace. I must have used or seen Linux used on at least 30 laptops with 2 (or more) screens without issue in the last ~10 years, counting the ones at my previous employer as one because they were all the same hardware even if people ran different distributions.