What do I need to look for when buying a new rear wheel?
The wheel's hub has one of three possible rotor mounts - there's 6 bolts, two sizes of centerlock, and a very small chance of something else weird.  


You need a hub that matches your rotor's mounts.  And yes there are adapters available, but you should buy what fits properly.  Do buy new bolts rather than reusing the old ones.


You have a 10 speed cassette, so you need a hub that takes a 10 speed cassette.  I believe 8-11 are all the same depth but the inter-cog spacing drops. 
You probably will get a new skewer with your new wheel.  They're relatively cheap, you probably want a 135mm long one for the rear.  Your bike does not suit through-axles so you require QR-based wheels, or wheels that have adapters for QRs.  Wearing away like this might be a sign of something else being wrong.
Yes, but it gets messy very fast.  Refer to the large collection of existing Q&amp;A on this:


Can I combine different wheel sizes in one mtb?
Is it worth swapping MTB front forks/wheel to ISO 584/27.5&quot; from ISO559/26&quot;?
Can I change my 28&quot; wheels for 24&quot;?
Disadvantages of run 650B wheelset on a 29&#39;er
There are a lot of questions with very good answers on this site - do search them out.