The receptacle manufacturer should document how many wires can be put where. In general, if there are screws, you can use at most one wire per screw. For quickwire/backwire holes, you can only use one wire per hole, and further, that one wire can only be 14 gauge. Previously, 12 and 14 gauge were allowed for quickwiring, but not any more. The only kind of receptacle where you can directly attach a whole bunch of wires are those where the screws tighten down a bar onto another bar. Then, you stick stripped wires between the two bars and tighten the top one down to clamp them between the two bars; you can run quite a few through there.
Another limiting factor is box sizes. Boxes have a fixed wiring volume, and there are rules for working out the minimum volume box needed to accommodate a given number of wires, devices, and clamps. Assuming you're working with 14 AWG grounded cable, I'd expect the box to accommodate 4 hot + 4 neutral + 1 device + ground wires, so 10 wire volumes, and 14 AWG is 2 cubic inches per wire, so you'd need a 20 cubic inch box. That's a bit on the large size for a box holding a single receptacle, but there are boxes and box configurations (box + mud ring) that would accommodate that many wires, so it's not unreasonable.
As for wiring everything together, you should pigtail everything and then hook up the pigtails to the receptacle. The pigtail presents a single wire for slipping under the screw terminal. Remember to wind the J-hook you'll slip around the screw in the direction that you'll turn the screw to tighten it, so down on the left side and up on the right side.
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Usually they do require the cover plate to verify the box is set correctly. Each inspector is different but some don't like DIY folks, especially in Lane County, Oregon. Benton & Linn counties were not so bad (I haven't worked up in Portland for ~15 years) but they usually want everything finished. I have seen inspectors fail an install because the wires were coated with sheet rock mud & paint.