Road trip in the US, I have some questions. Max 4k on a used car.

Firstly, me and my 3 mates are planning a road trip from New York to Los Angeles, and we would very much like, if its possible, to buy a car instead of renting one, but after some research, we find out that Insurance in mandatory. Is this also the case if we are only owning this car in 1 month maximum.

- Is insurance mandtory, even though you are only going to own the car for 1 month maximum?

- What places should we visit on the trip to LA?

- Is there anything we need to know about buying cars in the US(NY) that could take s by surprise?

- We are planning on using maximum 4k dollars on a used car, exclusive the mandatory insurance(if this is the case). I have been looking on the used Crown Vics, but if you have better advices, I would be glad if you commented, and linked me. It wouldn't hurt if it had some power under the hood.

- We are on a tight budget, so we are planning on sleeping in the car or at cheap motels most of the time, or even buying a tent, although this might be illegal.