Can I stay overnight to get some night sky shoot?

Yes

Absolutely. Book early if you want to get a spot

Yeah.you can

Yes they do have camping sites

Yeah

There are specific areas you can lodge/camp, so just check the info on wherever you want to look from. Their nps website is really helpful

There are overlooks on Skyline Drive for awesome overnight shoots, but be sure to check for fog. Around Mike Post 4 is the Shenandoah Valley Over look heading south.

Yes, you can