In high school, I bought my violin and bow used for around $750. While it was fine for orchestral/symphonic stuff, it definitely paled in tone, volume, and responsiveness compared to some of my friends' more expensive violins.
When I got to be the soloist during my senior year, I borrowed a much nicer violin from my director and bought a new bow because I simply couldn't get the right sound out of my own violin.
1) The operative pieces of wood here are the nut, the bridge, and the fingerboard. These pieces are all non-integral, relatively easily, cheaply replaceable parts of the violin. I can see how a nut or bridge that's too high would make the violin harder to play with good intonation, and for sure, a fingerboard that is worn or misshapen could, but other than that, I don't think there's any other part piece of wood that matters.
2) Not a rule, but sort of a default rule of thumb for a certain level of violin playing is don't play on open strings because you can't do vibrato and you can't adjust the pitch while you're playing. I remember my kid's intense, Teutonic teacher saying to him, "Well, your strings are out of tune, but that's NO EXCUSE for playing out of tune!", meaning, she expected my kid to have been able to make the adjustments on the fly with his finger placements.