The card says: If the runner is tagged, dedicated response team gains "Whenever a successful run ends, do 2 meat damage."

In timing structure of a run rulebook, 4.4 "when successful" trigger their conditions.

Does this means the ability triggers even after the runner successfully end with accessed and trashed this card?

Dedicated Response Team's trigger is "when a (successful) run ends", not "when a run is successful". A run ends after the runner is done accessing (and trashing, if applicable).

So, if the runner hit Snare! while DRT is rezed, that is 5 damage?

Yes (unless the runner avoids the tag, of course). Since both Snare! and DRT are assets, the only way for the runner to access a Snare! while DRT is active is for them to be on different servers, so the runner cannot trash DRT in the same run he accesses Snare!. Note that the Corp must have rezzed DRT before the runner accesses any cards.

What if you have two Dedicated Response Teams installed and rezzed in two different servers. if the runner ends a successful run tagged, does he take 4 damage?

JamesBond said:

yes. they are non-unique, and, therefore would stack, because each one triggers off the same successful run ending.