Do you have to provide a permit/evaluation from a doctor and a psychologist like in Poland?
No, but that would prevent a lot of sick minded people from owning guns. Such a law would never pass though. There's way too many gun owners and a very powerful lobby.
I understand there are different laws in different states?
Yes, some states for example allow private party sales without any sort of checks besides showing an ID which is a big thing the federal government wants to clamp down on. I know most stores and private sellers are pretty good about sales and record keeping but not all. In the Chicago area there's hundreds of gun stores, but an investigation found that only 4 or 5 were the ones really driving all the gun sales that ended up being used in crimes.
In some states you don't need a license to buy a shotgun or rifle, but you do a handgun (which makes no sense to me personally), in others you don't need a license to buy all 3, in others you need a license to buy all 3.
Some states leave disrection up to state police, other's don't. Some require you to show "good cause", other's don't.
You go across state lines and the laws can be vastly different for things like firearms, marijuana, gambling, etc. Here for example there's slot machines at every other gas station and you can buy marijuana legally, but gun laws are fairly strict. You go 20 miles west and gambling is restricted to "riverboats" (basically a barge in a shallow river), you can get years for possessing a couple hundred bucks worth of weed, but gun laws are pretty lax.