You could trouble shoot the rear by hooking up the rear caliper to the front lever/hose. And then the rear lever/hose to the front caliper. Time consuming but sine you know the front works doing this will give you an idea of what part of the rear is bad. I'm gonna guess there is a bad seal somewhere. There are the rebuild kits for the lever/master cylinger and the caliper. Once you diagnose you could rebuild that that section or just do all. I have bought my rebuild kits from universalcycles.com.

If I were if your shoes. I would try to diagnose as I mentioned above. Then check on cost of parts vs buying new or buying used

For your other question. You can run a new brake system with your current rotor. But might need a new caliper mount depending on what brand brake you buy.