The mode 3/a drops occur on SkyBeacon when not being interrogated by radar- a known issue. You probably flew out of coverage from a radar. The low percentages of the errors (integrity and accuracy - NIC and NACp) shouldn't be a concern. I don't know about the SkyBeacon adjustment for the air...
The N-number the ADS-B broadcasts is used to link the track on the controllers scope, with the track of your aircraft. Up until ADS-B, that's what the squawk code was for - to link the track with the aircraft. ADS-B gives a secondary way to do that - the N-number.
Flight-aware will not tell you whether the equipment is working properly. You really need to check a PAPR. Also, ATC does use the ADS-B output everywhere in the U.S. - it's just not obvious to the controller. That's by design. Controllers control - not engineer or troubleshoot.