The MSP checks 16 of it's databases as well as being the POC for the NICS.
Yes, the No-fly list is one of them.
Ummmmmm, No it's not, since the FBI does not share the no fly list with the NICS database or any outside organizations that are not a part of the security group that utilizes and updates the no fly list.
The no fly list was a major bill last year as well and all of this came out in testimony.