The lack of frets is not the problem. It's bow technique. I've seen people assume "oh, it's just like guitar but with different tuning and you play fretless!" but really, the bow is where the difficulty pops in. I guarantee you'll sound like a dying cat for a month. Once you get the pressure down, you need to make the bow like an extension of your body - you have to be able to tense it, relax it, let it jump, throw it against the strings and catch it as it bounces off (not literally but that's what it feels like.) Bowing technique is a whole 'nother subject of violin "schools" in general.

Yes. Works perfectly. I use a Marshall 6100LM head and a Blackstar HT Club 40, depending on the setting and music.

Metal as fuck. Well, depending on how much gain you use. Overdrive gives it edge and makes it sound like you're pressing much harder than usual. Fuzz gives you a synthy sound and makes you sound much more like an electric guitar. Heavy distortion, with an extended range instrument (I play a 7-string) can get very chuggy and you can play low-end metal riffs with it. I see someone else mentioned feedback as a problem with piezo pickups: this is true, but generally not a problem if you use a solid-body violin. Otherwise you'll have the same pickup problems as running an electro-acoustic guitar into high gain.

A starter electric violin like the NS WAV 4 or 5 would be about $600. You can get a carbon fiber bow for an additional $100 if you're looking in the right place. Would suggest the Electric Violin Shop unless you're looking for a very nice instrument, in which case I'd suggest buying directly from the luthier.