I've got a Jay Turser LP replica and I'm running it through a Boss DS-1 into an Epicrap Studio 10w. Whenever I try to play certain strings (most notably the 4th string, 9th fret) and I take my finger off the fret I get all sorts of noise and the faint sound of a lower note ringing through my amp. (i'm not accidentally pulling off because it's not as low as the open 4th string) Also, my 6th string vibrates a lot when I'm playing, adding to the noise and feedback.

I haven't had the thing professionally set up yet, but I'm thinking that might be necessary. It's already frustrating enough trying to get a decent metal tone from this awful Squier bundle amp and single coils. Any ideas?
Sounds like one of your pickups could be too high.
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hmm, could be. My pickups are really strange looking. My bridge pickup looks really high, but I don't know enough to say for sure. I'll just take it to the store.