1) my FC has done lot of clear runs to some member in unsynced so I'd assume it's possible. It of course still requires them to be at least of the minimum level for the content and have it unlocked.

2) what quests you mean? Many quests in the game have the sort of mini instance fights and they are always solo only. Can't really start listing them, maybe look in some xiv wiki. However if you mean Leve quest, all grand company Leve quests need to be done on solo (this used to hold for all Leve quests before HW I think)

Also for your original post 3) question, no, there is no way to do Fates unless you sync to the level. However if you want to help friend doing them, while you can't hit the fate mobs if you are not level synced, you can still heal people who are doing the fate while staying on your original level. Of course that means you won't get the fate done on yourself.