I have some rust on my fixie and I need help stopping/removing it, so I have two questions:

The first is, I need something I can use to remove the rust that has already started to set in. Is there a product or home remedy for this?

Also, there are mounting holes on the bike and I can see that a bit of rust has started to grow inside them. If I plug up the holes with an epoxy or putty of some sort will that stop the spread of the rust because it has no oxygen getting to it?

Search the forums for oxalic acid bath to remove rust. Interior of frame can be treated with Framesaver or Boeshield. Good luck. And no, I wouldn't plug holes, but you can put SS bolts in any threaded holes to prevent water getting in.

I honestly only get a little rust in my fork and frame drop outs. I always sand it off a little bit there and spray a spray paint can clear coat on it once in a while. But then again all of my bikes are used as commuters, not show bikes so I don't really give a fug. This has a whole section about rust on frames from the legend himself. Bicycle Frame Refinishing

Naval Jelly and fine Steel Wool fix a lot of evils.
I have a *****/*** protectant product called "Tuf-Glide" that works really well to prevent rust coming back.

Surface rust is nothing to be afraid of assuming you're not storing it long term outside exposed to the elements.
If you're worried about the rust eating through the tubing that'll take decades.