No, as its a communications licence and all transmissions must be directed to another amateur, which your UAV certainly is not.
I've never seen that restriction.
My understanding is that it is a license to experiment with radio waves.
Depends. In Slovakia where I am from it is still illegal for HAMs to use packet radio nodes that are connected to the internet, for example - even if it is only for a HAM to HAM connection. The reasoning goes that as it is going over non-HAM network, devices on that network will control the transmitter (indirectly by sending data packets) and as they (or rather their operators) are not licensed, it is illegal, period. So even stuff like Echolink is out.
Our licensing authority is seriously backwards and hyper conservative, they were among the last countries in the world to drop the requirement for Morse if you wanted to have HF privileges. And even now the basic N class license is severely restricted on HF, basically allowing you to use only the Morse parts of the bands, sans anything remotely usable for DX-ing (aka - "you didn't want to learn Morse? OK, we cannot drag our feet anymore because CEPT, so here you are. But screw you, no SSB for you, buster!").
Running video over a HAM band would be likely a no-no as well, because it isn't an approved mode, as would be operating an unattended station (a drone). Unattended operation (repeaters, packet nodes, etc) requires a special permit.
My guess is that it is the heritage of the communism where owning a radio transmitter (even a CB station!) required that you (and all your relatives!) were considered "reliable" by the communist party, were vetted by the secret police and it was a privilege of a few, certainly not a right like today where you pass an exam and get your ticket automatically. Unfortunately a lot of the "cadres" of that era are still sitting in the same offices.
In contrast to that, in Denmark where I lived before I have got an all-band/all-mode HF license up to 100W power with no questions asked, it was enough to show my Slovak ticket. In France where I am now it is likely similar.
So to correct your understanding - it is a license to operate an amateur radio service station using approved frequencies, modes and power levels, not to experiment with radio waves. Or at least it isn't the universal understanding of it. The experimentation certainly can fit into it, but the legal frameworks could be very restrictive, depending on country.
And anyway, HAM license doesn't cover operation of commercial equipment in the ISM bands. I didn't check how much overlap is there between normal 2.4GHz wifi band and HAM allocation there - there it could be potentially possible, but one has to carefully check the local rules about interference and modes allowed.