Short answer: usually to your own next car, rarely to someone else, and never to another state. The details depend on your state.
Usually yes. Most states let you move your personalized plate to a different vehicle registered to you in the same state, typically with a form and a transfer fee. Check your state's official page for how it works there.
In many states, a vanity plate stays with you rather than the car, so you take it off before the sale and either move it to your next vehicle or turn it in. Rules differ, so confirm with your state before you hand over the keys.
Generally no. A vanity plate is tied to your registration, and most states don't let you give or sell it to someone else. A few states allow transfers in limited cases, such as to a family member, so ask your DMV if that matters to you.
No. Each state issues its own plates. If you move, you'd request the same text from your new state, and only if it's available there. Check whether your plate is open in your new state before you move.
In almost every US state, no. The registration belongs to the state, not to you, so there's no legal way to sell it. That's different from countries like the UK, where personalized registrations are bought and sold.
The plate eventually goes back to the state and can be issued to someone else, usually after a waiting period. Here's how that works, and how people catch plates the moment they come back.
In many states, yes - the plate stays with you and can go on your next vehicle. Check your state's rules.
Usually not. Most states don't allow it, though a few make exceptions such as transfers to family.