Yes. One easy place to see this is Alta (skiing only) and Snowbird (mixed use) side by side on the same mountain in Utah. I enjoy both and prefer skiing the snow and runs of Alta.
I'd prefer snowboarding Alta too, but seeing for myself the difference to the snow and to the traffic patterns, I can see my snowboard would undermine Alta's preferential characteristics, so I'm happy they're separate.
They do, and that grooming cost is indirectly covered by the lift ticket prices or the event sponsors.
But the run will still be closed for a while, because it doesn't make sense to continually groom the run leading up to and during competition or practice. Generally it's not closed to just snowboarders, but to anybody who isn't involved with the event. At least that's how they do it here in CO.
[0] https://www.youtube.com/watch?v=XPZDEWBzneY