So, the a/c is a must for my car?

NO, it just means you won't have cold air. But I would think in AZ that would be a necessity.

Yes, for all of the functions of your car to operate correctly you will need to have the air conditioning system repaired. If you can live without the A/C and are not worried about the defroster not working correctly you can leave things as they are. Don't forget the remote possibility of the belt breaking and causing other damage.

yes if you run your ac compressor and it breaks it can cause the cost of the repair to go up.. yes you can also run it with the ac off :) no defrost ...