Any non-copy effects that applied to the creature Assimilation Aegis is attached to before it becomes a copy of another card will continue to apply once it becomes a copy. The same is true of any counters that are on that creature.

If Assimilation Aegis leaves the battlefield before its first ability resolves, the target creature won't be exiled at all.

If there are no creature cards exiled with Assimilation Aegis when its second ability resolves, the creature Assimilation Aegis is attached to won't become a copy of anything else. It will remain just as it was.

If you somehow exile a noncreature card with Assimilation Aegis's first ability (possibly because it was a copy of a creature while on the battlefield), you can't choose to have the creature Assimilation Aegis is attached to become a copy of that card when Assimilation Aegis's second ability resolves.

If you somehow exile multiple creature cards with the same Assimilation Aegis (probably because its first ability was copied or something caused it to trigger more than once), you choose one of the exiled creature cards each time its second triggered ability resolves. You don't have to choose the same one each time.

When Assimilation Aegis's second ability resolves, the creature Assimilation Aegis is attached to copies exactly what was printed on the original creature card and nothing else. It doesn't copy any information about the object the card was before it was exiled.

When the creature Assimilation Aegis is attached to becomes a copy of an exiled creature card, it's neither entering nor leaving the battlefield. Any enters-the-battlefield or leaves-the-battlefield abilities won't trigger.