Shower – How to install a drain cover in the shower for this opening in the floor

Go with your second approach of getting the right size fitting except don't bend the pipe, just deal with the angle. You pan lip should be higher than the ending (with fittings) top of the drain pipe so it can drain but that's all you need.

Why? PVC fumes are extremely toxic and you have a significantly better chance breaking the pipe than fixing the angle; then you'll have to tear up the floor to fix the pipe. A washer machine drain pan is used for worst case scenarios, not day-to-day use so as long as water can drain you'll be fine and a slight angle may actually decrease the minimum height of the drain anyways.

As for drilling the hole in the pan, mark it in place and drill it somewhere else with a sacrifice piece of wood/etc. underneath. You don't want to hold it up or anything because you'll need a decent amount of pressure to hold a bit in place and plastics like to fracture if not properly supported.

The plumber's putty will not work, pull it all out of there please. The pop-up assy. should have included a fairly thick tapered rubber gasket to seal against the underside of the drain hole. This gasket has a flat back for the "crappy plastic" washers to rest against, those are actually friction rings designed to let the nut spin freely and press up against the rubber cone washer. I think you are missing the rubber washer.

Plumber's putty should only be used to effect the seal between the upper drain trim ring and sink.