Re: Life of the Party
You get the Life of the Party reward when you do a certain number of actions during a party--or, more specifically, when you spend your party points. I've noticed I usually get the reward after I use 8 or so party points if there are only low level actions, but I've gotten it after 6 points spent if I'm doing all high level actions (like actions that yield 75 stars, like from a Christmas tree, a bar, or a bubble machine).
Your sim appears at the party when you bring them, but will only stay at the party if that sim does at least one action. If your sim doesn't participate while at the party, then they won't be visible to other players when you leave the party and so they won't be able to receive stickers.
Good questions, @UnicornFaerie000 , I hope I explained the answers clearly.