#1
I'm finding when I try to tap notes,especially when they're farther away, tapped note comes out dissonat. It sounds like a minor second or something. How can I clean this up?
#2
Sometimes you'll get the string between the tapped note and the fretted note below it ringing out, if that's what you mean. It's not a big deal since it's nowhere near the pickups.
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#3
^ That is the source of the noise.

This means you have to mute the part of the string below the note your tapping with your fretting hand, for example check out how the muting is done in this video:

https://www.youtube.com/watch?v=SIeYIKr0mwk

Ignoring the sound is usually a bad idea since even though the noise isn't as near the pickups as the other side of the string vibrating, it will still sound.