In my experience, a master student's final project typically constitute their first foray into / encounter with academic research. So yes, untrained in that sense.
The fact that you are paid doesn't preclude it being a traineeship.
No not really. A PhD student at most receives a scholarship, and currently at least in Europe has to undergo a curricular part comprised of a dozen or so courses. How does that not count as schooling?
If I'm not mistaken the only exception to the schooling requirement is if a student is able to publish a few research papers on a topic that are coherent and related to a concrete research topic that could support their thesis.