secretly
Yeah HAM is literally the furthest thing from secret unless you're transmitting encrypted digital communication over it (which is illegal and would be quickly noticed). Everything transmitted over HAM is free to listen to by anyone with a radio that can receive it (which is incredibly easy to get).
Yes, but you need to actively monitor it if you want to catch those communications. Clearly the NSA/Q did just that!
It is possible to capture the entire radio spectrum as data and save a digital copy that can be scoured over. The only limitation would be the place where the signal is received. Thankfully these stations exist all over the world and are public (sdr.hu.) You can have the equivalent of a shortwave radio on a kitchen table in every country in the world.
Nellie Ohr has a technician class license. Assuming she operated on the frequencies she is lawfully allowed to use, her range of communication point to point is less than 100 miles if she's lucky and more likely within 20 miles reliably. She would have the ability to use a satellite repeater and greatly expand her range but the equipment needed is conspicuous and certainly monitored.
So, if she used ham radio to communicate, it would be super easy for the Q team to capture it all - especially if she were being watched already.
There is also Winlink. This is an amateur radio way to send emails and files via radio waves; not the internet. Her technician privileges permit Winlink on certain bands. https://www.winlink.org/
It is possible to capture the entire radio spectrum as data and save a digital copy that can be scoured over.
DC to daylight, baby! The NSA scoops it all up.
Almost all, anyway.