Tonight, sleep legally

It depends on the state, and the range is wide. Some states allow a stay measured in hours that comfortably covers a night's sleep. Some cap you at two or three hours. Some prohibit overnight parking outright. There is no national rule: rest areas are operated by each state's department of transportation and each one sets its own policy. Check your state before you count on it.

A great deal, and it is the distinction most people miss. Many states prohibit camping at rest areas, meaning tents, awnings, chairs, cooking, and setting up, while still allowing you to sleep in a legally parked vehicle for some number of hours. Those states are telling you to rest, not to stay. If a state prohibits overnight parking, that is a different and stricter rule. Read which one the sign actually says.

Because they exist to stop tired drivers crashing, not to provide free lodging. The limits keep spaces turning over for the people the facility is for, and they exist partly because interstate rest areas are federally funded and states are restricted in how they may be used. That history explains most of the rules that otherwise look arbitrary.

Usually somebody knocks on the window and asks you to move, which is the common outcome and not a big deal. It can escalate to a citation, and in some states to a tow, particularly for a vehicle left unattended past the limit. The realistic cost is that you get woken up at 2am and have to drive somewhere while tired, which is the exact thing rest areas exist to prevent.

Usually, yes. A rest area is for a few hours of sleep mid-drive. If you want to actually stop for the night, dispersed camping on public land is free, legal, quiet, and frequently twenty minutes off the same exit. Rest areas are the right answer when you are tired now and the alternative is another hour of driving.