I recently decided to get into QIII, and downloaded the demos on Steam. The regular Arena works fine, but the Team Arena reaches fullscreen, then minimizes and crashes. I'm not sure if it's because I had the demos only, but I googled it and found nothing. Does anyone have any advice?

Any error messages?
Are you loading the mod through Q3A game or using a stand alone .exe?
Also, which patch are you using?

1. It starts up normally, with the console showing up for a second or 2 and then it switches to fullscreen. No error messages besides "taquake3.exe has stopped working" from windows itself. Note that it can't even get to the main menu.
2. Not sure what you mean by loading it through the Q3A game, but I assume I'm using a stand alone .exe, seeing as it has it's on separate directory in the steamapps folder.
3. I'm using no patch, it's a demo and regular Arena works like a charm. I tried looking around for a patch for Team Arena, but I wasn't sure what to download.

I am also running on Windows 10, could that be an issue?

Can you try running it in compatibility mode?
Team Arena you downloaded from steam as a demo?

Yeah, I'll try. And I did download it from Steam as a demo, since I'm planning on buying QIII soon.

BTW, what settings do I use for compatibility mode? It's preset for Windows XP automatically

I use Win7, but you should have no problems with XP.
Nobody really plays Team Arena.
You're better off with Quake 3 Arena, Quake Live, CPMA, Defrag.

Alright. Still no results.

The only reason I bothered with it was because base QIII didn't have CTF and all that, but as long as that's in the other games, I guess I can survive.

Regular Quake3 has CTF. You just have to find the right servers which host that gameplay mode.

Hey, I don't know if you ever solved this... but it'll work if you turn off the Steam overlay.

I never solved it directly... but when I eventually got the QIII Arena pack, Team Arena actually started up, from the Steam Menu and from base QIII. I think it was matter of the demo not working, but the actual game working ($$$).

Also, never needed to turn off the Steam Overlay for the purchased versions.