With most any vintage cassette deck, you want to start by treating the record/play switch with a contact cleaner. Many people swear by the DeOxit brand. The R/P switch that I'm referring to will be mounted on the internal circuit board. It's a long, multi-terminal component, pretty easy to spot.
Cleaning the R/P switch is very likely to solve the right channel cutting in and out. It's possible it will solve the noise, too, but if the noise is coming from elsewhere, it will be more difficult to track down.
The one KX-1030 that I can remember working had a playback noise problem, too. And it was indeed difficult to track down. But what was really memorable about it was that the source of the noise was a capacitor. (I've noticed that people often finger caps as likely suspects in noise problems, but in my experience it's rarer than hen's teeth.) I'm afraid I can't remember enough detail to say where to find the corresponding cap in your deck, but I vaguely recollect that it was a film type, in a "green blob" package.
Good luck,
chazix