1. You don't have to fly out there, but if you want to use your YT-1300, you need to either fly there or have someone get it for you.
2. If you own the Y-Wing, it'll stay at the location of the YT-1300, since you can only fly one ship at a time, and neither of those has a docking bay.
3/4. No. You can have some pilot NPCs fly other ships, if those are a fighter-class ship and of the same type as the one you're flying (you can have up to 11 pilot NPCs, which are automatically created, follow you). This is also available with a few select freighters.