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No need to drill out the holes. They should be sized appropriately already.
Also no need for a cutting / cooling / transport fluid -- The plastic doesn't really heat up enough to warrant that and I am unsure whether those fluids would weaken the plastic or not. When working with metal, the friction and forces involved can cause the tool to dull and weaken producing even more heat and eventually breaking the tool. It also makes the machining rougher and not as precise. Plastic is softer so you won't damage the tool, but you can build up enough heat to start melting the plastic. This is what may have happened to you. Go slowly -- if you're using a drill, try it by hand. It can be done with a drill, but it takes a very steady hand to keep it aligned with the hole and moving as slowly as necessary.
Yes to the half turn in and quarter turn back. The tap will have flutes cut into it (probably either three or four) which collect the cut material. Doing that quarter turn back helps collect that material so it doesn't collect in the path of the cutting edge of the tap. Not doing this can also cause additional heat or the collecting plastic chips can destroy the threads you are cutting.
One tap for each size should be fine as long as you have the correct diameter and thread pitch for the screws you are using.
I haven't tried self-tapping or tap-cutting screws. I'm not sure how well they would work. They would likely need the same treatment as the tap. You'd also need to find screws that fit into the same holes since these are all sized for metric screws. Something like this: https://www.mcmaster.com/#tapping-screws/=1bgsm3a. This is admittedly one of the tough things about this build. I didn't really want to use wood or sheet metal screws on it because they would require altering the design (I wanted to be as authentic as possible) and because they just don't offer the same aesthetic.
Some additional tips: Keep that tap as aligned as you can with the hole. Putting it in at an angle to the hole will cause problems. The holes around the side of the main casting don't go straight towards the center, they are offset at an angle (they follow the same angle as the slots cut in the flanges of the main casting). Take your time, examine the hole, align the tap, turn slowly, and do the quarter reverse turns every half turn. I tend to not do the reverse turns until the tap is nice and established in the hole and I like to re-examine the angle I'm at after a couple of turns to ensure I am still aligned properly.