I'm traveling across the country and want to park my van overnight and sleep for one night, am I permitted?

Yes u have 48 hours then you must go that's where majority of parking lots.

Yes, at all Walmarts it is legal to park for 24 hours in there PL

You have 24 to 48hr to park unless they have signs posted at facility

Yes

Try calling and asking. They might appreciate that and give you permission.

Not sure. But you wouldn't be the first to do it. Also try wawa parking lots.