If multiple creatures enter under your control at the same time, Renegade Doppelganger's ability will trigger once for each of them. You choose the order that those abilities will resolve. It's possible to have Renegade Doppelganger become a copy of something, activate an activated ability of that creature, then have it become a copy of something else under this scenario. The last ability that resolves that you choose to use determines what it ends up a copy of.

If a creature enters under your control and Renegade Doppelganger's ability triggers, it can become a copy of that creature when the ability resolves even if that creature has left the battlefield by then. If it has, its last existence on the battlefield is checked to see what it was (specifically, if it was itself or if it was copying something else).

Noncopy effects that have already applied to Renegade Doppelganger will continue to apply to it after it becomes a copy of something else. For example, if Giant Growth had given it +3/+3 earlier in the turn, then it becomes a copy of Runeclaw Bear (a 2/2 creature), it will be a 5/5 Runeclaw Bear.

If Renegade Doppelganger becomes a copy of a creature that is itself copying something else (for example, if the creature that entered is a Clone), then Renegade Doppelganger becomes a copy of whatever that creature is copying.

If the creature Renegade Doppelganger copies has an ability that triggers at the beginning of the end step, that ability will trigger and resolve before Renegade Doppelganger's copy effect wears off.

If Renegade Doppelganger becomes a copy of a token, Renegade Doppelganger copies the original characteristics of that token as stated by the effect that put it onto the battlefield. Renegade Doppelganger doesn't become a token.

If Renegade Doppelganger becomes a copy of a creature with an ability that causes you to make a choice (such as naming a card or choosing a color) as it enters, and another ability that refers to that choice, those abilities of the Doppelganger won't do anything. You won't have the chance to make a choice since the Doppelganger is already on the battlefield, so the value of that choice is undefined.

During the cleanup step, Renegade Doppelganger's copy effect wears off. It will go back to being a 0/1 blue creature named Renegade Doppelganger, and will have its original ability again. Note that this happens at the same time that damage marked on the Renegade Doppelganger is removed. If it was dealt damage earlier in the turn, this damage will not cause Renegade Doppelganger to be destroyed when its copy effect wears off.

When a creature enters under your control, you can choose not to copy it. In that case, Renegade Doppelganger remains 0/1 and still has its triggered ability.

If Renegade Doppelganger becomes a copy of another creature, that doesn't count as having a creature enter. Renegade Doppelganger was already on the battlefield; it only changed its characteristics. If Renegade Doppelganger gains any "enters" triggered abilities, they won't do anything. The same is true of any "as [this creature] enters" or "[this creature] enters with" abilities.

Renegade Doppelganger's ability isn't targeted. It can copy a creature with shroud or protection.

If Renegade Doppelganger and another creature enter under your control at the same time, Renegade Doppelganger's ability will trigger. When the ability resolves, Renegade Doppelganger can become a copy of that creature.

If Renegade Doppelganger becomes a copy of another creature, it copies exactly what was printed on that card and nothing more (unless that card is copying something else or it's a token; see below). It doesn't copy whether the original creature is tapped or untapped, whether it has any counters on it or Auras attached to it, or whether it's been affected by any noncopy effects that changed its power, toughness, types, color, or so on.

If Renegade Doppelganger becomes a copy of a legendary creature, you will have to put one of them into the graveyard as a state-based action.