Is there an easy way to check if everything is set up properly and also to get tips on what needs to be changed?

Not really, but the most common mistake is to set TERM in your shell startup files (.bashrc, for example). If you do that, then you could end up setting it to a value not supported by your terminal emulator. If you don't set it, then your terminal emulator will set it to an appropriate value for you.

I thought one needed Ctrl+L to refresh the screen when it gets corrupted by program output in "gdb -tui".

That's absolutely true, I'd forgotten about that. It's hard to tell from the screenshot if that's what happened, but it's quite likely.