The 9v battery question is more complicated…
Connected to Vin, there will be regulated 5v to the avr until the battery reaches about 7 to 8v (depending on exactly which regulator chip your board has.) But the avr will run at less than 5v, so in theory if the regulator supplies Vin-2V, the avr might run ok until the battery is down around 5 to 6 (avr min voltage is something like 1.8V, but not at 16MHz.)
A 9V battery outputting 7V is pretty much “dead”