Not all their planes are lie flat. If you tell us your planned itin, we can help you find the best, but generally, look for the aircraft type to be 77W, that will be your best bet.

Yes, you can use F lounges.

Funny you mention Virgin Atlantic as being the only three class product you've been on. Seeing as it's "Upper Class" is no different than BusinessFirst or BusinessElite or any other Business class you're probably used to. Singapore's First Class is in a very different league.

Between Hong Kong and Seoul...Hong Kong every time.

I'd skip Dubai; you'll end up on non-lie-flat planes and that would suck. I'd go San Fran - HK (stop) - SIN (stop) - Moscow (stop) - Houston (connect to SFO). The San Fran - HK and the SIN - Moscow and Moscow - Houston will all definitely be on 77Ws, so just watch out to make sure you get a 77W between Hong Kong and Singapore - and you'll be golden.

Singapore's First Class is (usually) a great experience. Just be forewarned that you'll never look at travelling back of First Class the same again.