If the Runner is tagged and makes a successful run, both and trigger simultaneously. The Runner, being the current player, has to make another run before taking the 2 meat damage. If the runner makes another successful run they will trigger Dedicated Response Team again taking 2 more meat mamage. However, if the second run is to trash the Dedicated Response Team, then the Runner takes 0 meat damage after the run ends.
The runner makes a successful run. At the end of this run, both Doppelgänger and Dedicated Response Team trigger. Doppelgänger must trigger first, because the runner is the active player, and the runner chooses to make another run. When will Dedicated Response Team deal damage?