So I'm looking at a rental online and it has shingle siding.

Between the two of you you should have a reliable roofing contractor to evaluate the roof on a full gut or just a re-shingling. 

The 1135 sq ft brick house needs $10K repairs that would include but not be limited to new shingles and a new AC and maybe kitchen appliances.

Arizona allows you to put on a second layer of shingles without pulling permits.

They are going to remove shingles on valleys and low slope of roof, install weather shield stuff, and re-shingle.

EFIS siding or Asbestos shingles or siding8.

There was some patch work done and the original contractor who fixed this made no effort to match the shingles.

I did a re-shingle on a duplex in 2005 and the roof completely failed last year. 

The homes all have new metal roofs or shingles that have been done in the last 4 years.

Roof size approx 2700sq feet- gaf woodland lifetime designer shingles (gives the slate look similar to what's on the house) - If I choose plain architectural shingles, I'll save 2k- TPO PVC flat roof over 10x30 extension- soffit & fascia repair and replace where necessary (some of the soffit wood has gotten rotten and would need to be replaced)- chose copper valley instead of aluminum (said I'll save $500 if going with aluminium)- plywood mostly in very good condition because of great original quality of the wood (said I'll probably only need 3 sheets replaced)- replace downspouts where necessary, maybe he'll replace 1-2- install cobra ridge vent