You just need to touch the PC, at first and then occasionally. You don't need to be properly earthed at any time, just not holding any excess static charge. (I have a mental image of you going across to touch the radiator, then walking back across carpet in your socks to do your PC build - which obviously you don't do or you would have killed something by now, but still...)
Don't fret about thermal paste. Decent paste is decent enough. Unless there's some unbiased testing that shows a marked difference, but any tests I've seen of the usual ones has barely a degree or two of difference.