Say I were to gift a friend a skin for a champion they don't have. Would it save the skin for them until they buy the champion? Or would it just not let you?

You can (should) try it on your own (for PBE sake).

My PBE experience: Rejected by the store with short notice that my friend does not have that champion.

Moreover, I received a notice some days later that my friends has got my gift, which I think that they only receive it/ you give it out successfully when they log-in.

I also ask myself what happens if you try to gift a friend a Champion they already have. Since all Champions are shown when you click on gifting a Champion.