Any recommendations on how to best cut the traces on the board? Scrape them with a nail? Cut them with utility/x-acto knife?
Xacto knife: two cuts, slightly apart, angled toward each other, almost making a mini V-groove. Then just peel the copper away from between the two cuts. A gap of a millimetre (or more) is the minimum needed.
Actually, the part I'm most worried about is soldering wire to points on the board where there aren't any existing pads (marked in green). Should I just scrape up a small area for the wire?
You can do it without any scraping, but for the red wires I personally would scrape away the green stuff to make room for new pads. That way I wouldn't be soldering too close to any components I might damage.

Cheers