When you ask another player sim to move in do they lose that sim.

No

Its should just be that sim you can control aen a part of the family

No. Other players do not lose said sim, your device just makes a copy of that sim.

the sim will still exist like you will see them around. but yes they will lose their sim. I just gave up one of my sims to see what would happen and it is basically like you never had it, all the career and relationship progress with the sim is gone and you can make a completely new sim and start over. when they get your sim I believe it also starts over and the only thing that remains is the relationship with their sim (like career progress and hobby progress is lost). seems kind of stupid honestly and is definitely one of my major complaints of the game

If you favourite a sim they will show up in your game.

If you ask another players sim to move in, the player will be given the option to keep their sim or let it move out (into your game) in exchange for a heirloom.

I was asked by another player and said no :)

Just an updated answer, when another Sim from a player decided to move in, that Sim will be automatically added to your roster and it will not be present on the other player.
Favoriting a Sim will show up in your game.

As far as I know, they will not lose the Sim.
You can socialize and you can see your favourites in your game.