Aboleth Spawn's ability cares only about triggered abilities of the creature that's entering, not abilities of other permanents that trigger when that creature enters the battlefield.

Triggered abilities use the words "when," "whenever," or "at." They're often written as "[Trigger condition],[effect]."

If the ability is modal (that is, it has a bulleted list of choices), the copy will have the same mode(s). You can't choose new ones.

Effects that modify how a creature enters the battlefield are not triggered abilities and are not affected by Aboleth Spawn's ability. These include (but aren't limited to) entering the battlefield as a copy of another permanent, entering the battlefield with counters, and entering the battlefield tapped.

Aboleth Spawn's ability will always go on the stack on top of the entering creature's triggered ability that caused it to trigger. The copy it creates will be created on the stack on top of the entering creature's triggered ability and the copy will resolve before that ability.

The copy will have the same targets as the ability it's copying unless you choose new ones. You may change any number of the targets, including all of them or none of them. If, for one of the targets, you can't choose a new legal target, then it remains unchanged (even if the current target is illegal).