Illusory Gains's last ability is mandatory. You can't choose to have Illusory Gains not move.

If Illusory Gains can't legally enchant the creature that enters the battlefield (perhaps because it has protection from blue), it remains where it is. You retain control of the creature it's currently enchanting.

The last ability of Illusory Gains doesn't target the new creature. It will become attached to that creature even if that creature has hexproof or shroud.

If multiple creatures enter the battlefield under an opponent's control at the same time, Illusory Gains will trigger for each of them. You may put these abilities on the stack in any order. As each resolves, you'll briefly gain control of the associated creature. The last one to resolve determines which one you'll ultimately control.