If you’re buying from an FFL anywhere in the US you absolutely are getting background checked. FFL’s don’t want to open themselves up to more liability than they need to.
The only way you’re not getting background checked is private sales from your cousin vinney you suspect is in a gang or your redneck uncle Rob.
That being said, background checks are different from waiting periods. They usually only take 5-10 minutes to get approval. You literally just awkwardly stand by the counter and dick around on your phone while you wait
Those are NFA items. Each one requires a $200 tax stamp (in addition to the sale price) plus you have to fill out a form and wait for it to be processed. Average times are now around 6 days but it used to be several months on average