Can I build a valid village (actually an iron golem farm) completely underground (no skylight access above)? If no, what is the closest solution to have an iron golem farm underground?
Not completely, no. You will need some sort of air shaft.
Villages, in Minecraft, are just a collection of doors that meet the condition of having more spaces on one side exposed to sunlight (This page on the Minecraft Wiki goes into the details). This means that a village/Iron Golem farm without any skylights won't work, as none of the doors placed will count as Village houses. The closest you can come will involve either open shafts, or else glass or something similar in the ceiling.
If you build the village completely from scratch it's a possibility. My villagers are still alive after a week of starting an underground village. You'll need a couple trees (learn via YouTube), beds for them to sleep (yes they still go by surface time) and plenty of light so zombies don't kill them. As for a golem I have yet to make one in mine so as I said it is a possibility. Now mojang generating one fully underground is unlikely but possible though the closest I've seen was the ravine, landbridge, or a village in the opening of a cave. I wish you luck on your building be creative and if you do build from scratch try making it as closely resembled as their home on the surface. A cat spawned in mine once the villagers got comfortable.
No, not without a glitch occurring. I was once playing minecraft on my Xbox. A village had spawned really close to a trench so some houses and a farm spawned there. A stronghold was right next to it. You'd be surprised how many villagers had wandered as deep into there as they were without getting killed by zombies.