The CRT is dangerous, yes, but discharging it is really easy. Take a flathead screwdriver, and an alligator clip, and attach one end to the screwdriver, and the other end to the "ears" of the CRT (which are connected to the grounding circuit in like 99 percent of CRTs). Slide it under the suction cup until you hear a POP and/or see an arc. If you don't, it might have already been discharged (most newer CRTs have bleeder resistors), but kinda wiggle it around regardless to make sure it's fully empty of any current. Now, the CRT is discharged and fully safe to touch anywhere to remove it. 😀
As for caps on the board, you can discharge them by shorting, but it's easier to just grip it away from their leads.
As for whitening the case, sanding or painting it may ruin it. I'm personally not a fan of retrobriting/whitening things, and I really only do it if the thing is seriously nasty yellow, but I understand if someone else wants their plastics to look nice; so, I would suggest trying to retrobrite it first before trying anything that could damage it further.