Answer to both is: "Depends".
Airplane mode
What is that nowadays, where air planes allow you to use WiFi and Bluetooth? Don't say this is an unrelated question in this context. Not only does the term originate there, but it can be as variable as how each airline deals with it:
$ adb shell settings get global airplane_mode_radios
cell,nfc,wimax,bluetooth,wifi
$ adb shell settings put global airplane_mode_radios "cell,nfc,wimax"
$ adb shell settings get global airplane_mode_radios
cell,nfc,wimax
What's that? The first is what you probably expected: all kind of signals are toggled by airplane mode (line 2). But it can be changed (line 3), and then looks very different (line 5). I've e.g. excluded WiFi on some of my devices. Different manufacturers/ROMs might have different presets. So there can't be a general answer unless you define "airplane mode". And of course there are apps that can automatically toggle all that (at least on rooted devices) – your "anti-theft" guess is a very good example for that. A paranoid person could even use e.g. tasker to configure a "watchdog": "Whenever my device leaves a configured geofence, automatically report its location to my server in 30s intervals. If needed, enable WiFi or mobile data for that." Might be complex, but doable. On non-rooted devices, that's limited though – not at last by:
$ adb shell settings get global airplane_mode_toggleable_radios
bluetooth,wifi,nfc
Again only an example, which might not be valid for each and every device (concerning line 2). You might already have guessed it:
adb shell settings put global airplane_mode_toggleable_radios "bluetooth,wifi,nfc,wimax,cell"
And with that, everything can be toggled again. According to a test performed by Prahlad Yeri, this even works on non-rooted devices. Settings applied this way survive a reboot (but of course not a factory-reset).
Any Android device which cannot transmit signals cannot be geotracked
Without some paranoia, I'd give a "yes" to that – though to be exact, the condition is hard to meet: as soon as it's powered up, it will emmit some "signature" which one could track – but that would need quite expensive equipment, and the range would be quite limited :)
If in this context you speak of an "attached GPS beacon": well, you could track a cake that way, or a cat, your cigarette box, or a brick for that – it no longer has anything to do with the device itself. Almost everything can be "bugged".