Gift a game to an american friend as a european

I play on the us servers sometimes with a friend and I would like to gift them the new call of duty (cold war) game.

When I’m on the us region and I want to gift a game, pick a friend etc. I see all my european friends but I don’t see my american friends for some reason. That friend is on my europe region friend list though which means I can pick them for the gift option.

If I gift them the game, they can play it in any region, right? I really don’t want to mess this up. We always play on the us servers.