licensed dealers are required to conduct background checks before conducting any sales.
No they are not.
They are required to have a purchaser fill out the appropriate forms to submit to law enforcement, who are then required to perform the background checks.
All that depends on State law where the buyer is a resident.
Some states allow cross state line purchases. Some don’t.
Some use a paper form submitted (physically or electronically) to state authorities for the background check. Some use NICS.
Some have a waiting period, some don’t.