Of course, it is not 'by' in the sense of 'past'. It is 'by' in the sense of 'near the mountain' or 'in the vicinity of the mountain'.
I was wondering about that... how would you translate the sentence with that notion of 'past', then?
Yes, but it also means 'past' (as in 'go past'). I don't know, perhaps there's a way to combine 'mellett' with an appropriate suffix on the noun.
No. You can only have a postposition or a rag-suffix, not both.
If you give the movement some direction, you're pretty much set. The most neutral approach would be "A repülőgép elrepül a hegy mellett."
I wrote onto because to me “at the mountain” would be an impending crash
I am not a native english speaker, but this sentence makes no sense to me. ...at the mountain. Frankly I am having a hard time to understand where this plane is flying.
at the mountain is not so good, then it seems like the plane is going to crash into the mountain.
by the mountain is much better, then it is like the plane is flying next to, over or above the mountain.
Unfortunately, as used here, the English "by" can suggest either (1) past or (2) near. Here no motion past is implied by the Hungarian sentence, so "near" is the intended meaning.