A pressure test using a air compressor could have detected the broken pipes.

All units should have overheating protection built into or part of the unit setup, wherein it simply delays the operation of the unit until it is safe for the unit to operate, and shuts it down as necessary in order to prevent damage to the compressor.

Ive seen many compressors fail at four to five years.

It is not the tenants fault(99% of the time) when the compressor goes out or a pipe breaks under the house. 

The words "fire hazard" got me concerned so I called an electrician who went out the same day to look at it.After looking at the problem, the electrician called me and said that the "clicking noise" she heard was the sound of the compressor kicking on in a dorm sized refrigerator she had.

@Peter Morgan I have heard of the AC compressors going bad due to running them in cold temperatures. 

Their quote:$2300 for a new condenser / all related installation$3000 for a new outdoor unit all in / all related installationIs there any reason not to buy a compressor online (looks like about $650 for the part) and have a general repair contractor install it for me?

For a small 3 bed 1 bath... if they are not too old, 1.5 or 2 ton HVAC (compressor and air handler) would be the most expensive part, right?

Act on that.You should also prepare a checklist for every system/appliance in the house that you go through for preventative maintenance between tenants (e.g. washer lint tube clean out, dryer lint house clean out, inspect gutters, fridge compressor check, window drafts/movement, etc. etc.).

He wants to use his compressor.