Do I need a special receptacle when more than the standard number of wires need to be connected?
If this is, in fact, a split outlet, you should be fine wire nutting the blacks and a black pigtail, the reds and a red pigtail, and you can connect the whites directly or wirenut them and a pigtail, as you prefer. You will need to break the tab connecting the receptacles on the hot side, at least (my guess is that they broke the hot tab and left the neutral tab connected, looking at the picture we're starting with.)

Connect that grounding wire to the new receptacle, as well...