[on2wheels52]

"Are there states in the US where someone who is not a US citizen or permanent resident etc. is able to legally purchase a firearm over the counter?"

The 4473 form has that provision. Question 11 (l) asks "Are you a nonimmigrant alien?"
Question 12 asks "If you are a nonimmiagrant alien, do you fall within any of the exceptions set forth in the instructions?"
Those exceptions are either a valid US hunting license/permit or a waiver from the Attorney General of the US.

"Which states are the slackest on these requirements?"

It seems we're all in the same boat, I suppose some states have additional requirements in addition to the federal ones.

"Could, for example, a tourist go into a gun shop and purchase a firearm, no questions asked?"

No. 20c (for the licensee) "All Aliens: Type and dates of documents that would establish 90-day residency."
20d "Nonimmigrant Aliens Must Provide: Type of documentation showing an exception to the nonimmigrant alien prohibition."

I can't imagine many firearms are transferred under these provisions. I know none of my 20,000 transfers have.
Jim