If you take the adopted trait, do you get a racial trait 'free' as a part of the adopted trait, or does the adopted trait just give you the option of making another trait pick a racial trait?
Adopted: I think it means that you get it for "free".
Half-elf: Yes, I think they count as elves for elf traits: They have elf blood, and often also live among elves.
As always, it's best to work out these things with your GM and make sure they fit your back story.