

The radios are the parts of your phone that communicate wirelessly. Most phones will only turn the cell radio off entirely during airplane mode, disabling mobile data does not typically turn that radio off.
Airplane mode should turn everything off (unless you re-enable things like WiFi, but that should still keep the cell radio off)
It shouldn’t be, no. But one of the big problems with phones currently is that the radio firmware is almost always a closed-source binary blob.
Airplane mode is probably better understood as the OS asking the radio nicely to not attempt to communicate with the outside world. The antenna is still there able to receive signals, and the radio technically doesn’t have to listen to the OS if it doesn’t want to.
It’s incredibly unlikely (researchers look for this kind of thing), so make sure your tin foil is on tight, but not impossible that a radio could store cell tower identifiers it has seen whilst on airplane mode and do something with them when it is allowed to communicate again. There’s also the possibility there’s some secret signal that can be sent to force a phone in airplane mode to respond.
Unless you’re up to some Edward Snowdon level stuff though, even if that last one exists, it’s probably not being used on you.