A: TLDR: If you play Get There First, the targeted Artifact goes from being in play in front of another person, to being in play in front of you. It does not go in your hand.

The (not really very) long version (with lots of parenthetical phrases):

When someone plays an Artifact from their hand to the table in front of them, this represents their having gone back in time to “collect” it (or capture it, in some cases!) So, when you (play) Get There First, it means you went back in time to just before they went back in time to, and collect the item yourself, so it would be as if YOU had been the one who collected the item (played it from your hand) instead of them, so it goes in front of you.