I've done the T2 to T4 transfer (had a UA cxl and they put me on US), and managed to do the whole thing in about 10 min, but I know the airport pretty well.

Not sure on luggage. UA/US should interline, but you'll want to check with F9.

As to getting between terminals, pretty much most of the airport-run shuttles can do it. When going from T2 Eastward (to T3 or T4), I would take either of the East Economy parking shuttles (Gecko or Roadrunner). The West Economy shuttle should go to T4 also, but not sure. To do the reverse, I think any should work (I've used both Gecko and Roadrunner from East Economy, past T4 to T2 or T3). When going from T4 to T3/T2, I'd probably go from the check-in level.

Lastly, there's a new bus that connects all of the terminals to the light rail. In theory, this should also work for you, but I'm not sure. By the same theory, the rental car shuttle should also work.

If you are getting on a shuttle at T2, ask which direction they are going. There is only 1 curb at T2, so all the shuttles will stop at the same place (close to the baggage claim). T3 & T4 have North & South curbs where shuttles on the North side head West, those on the South head East.

And YES you do have to exit/reclear security between terminals here at PHX. The exception would be within T4, there's ways to get from one concourse to another (ex US/BA to WN) within security. If you have *A status, there are elite security lines for both US and UA in their respective terminals.

My office is close to the airport, let me know if you'll have a long layover.