To answer your questions and to ask one...
1. Overnight Permits...I would not get one, I'd just leave before midnight on a day hike permit. The seasonal rangers will be sleeping like everyone else. However, if you feel more comfortable with an overnight, get one.
2. Smithsonian Hut...It's open and I have napped in it before walking out to the sunrise and cast Whitney shadow on the western horizon.
3. Campsites...I think 10 AM is the clear out time.
There is still snow on the mountain in Early June, which means at bare minimum Microspikes and trekking poles but more times than not crampons and ice axe and going up the chute rather than the trail. Are you prepared to use either?
