Yes. All you really need are paths. This is a bit of an overgeneralization since forests in different places have different growth habits, but many old growth forests have less underbrush. During the winter, these forests are pretty easy to navigate on Nordic skis. If he wants to skate instead of doing the classic stride, I’m sure he can afford to groom the paths. Grooming really just requires a fancy snowmobile.
It's not going to be entirely virgin since you need to actually clear some amount of paths to be able to lay down tracks, but you certainly don't need to clear it entirely. I'd say that cross-country ski tracks are compatible with forest conservation goals.
Potentially. Depending on the style of skiing, you may only need a 2-3 ft wide pathway. That said, ski paths tend to be cleared of trees/large rocks and span about 10-15 feet wide.
It might not get to remain a virgin old-growth forest. If the Swedish logging companies can get away with it, he probably can, too.
Who said it's a virgin old-growth forest?
I too doubt it is actually old-growth. Probably industrial forest. Which means that there is plenty of space around most of the trees.