As Aura Finesse resolves, if either target is illegal, the spell resolves but the Aura doesn’t move. You still draw a card. If both targets are illegal, Aura Finesse doesn’t resolve and you don’t draw a card.

You still control the Aura, even if you attach it to a creature you don’t control.

As Aura Finesse resolves, if the targeted Aura can’t legally enchant the targeted creature (for example, because the Aura is red and the creature has protection from red, or because the Aura has “enchant creature you control” and an opponent controls the creature), the spell resolves but the Aura doesn’t move. You still draw a card.

If you target an Aura that’s attached to a creature, you may also target the creature it’s attached to. Aura Finesse won’t cause the Aura to move, but you’ll still draw a card when it resolves.

You may target any Aura you control, and you may target any creature (including one you don’t control). Whether the Aura could legally enchant the creature is irrelevant as you’re choosing the targets.