Happy clauses and such are nice, but ask yourself these questions first: Are the systems that were failing near the end of their life cycles? Did the systems suffer from benign neglect (washer leak from lint clog, etc.). The air conditioner failure could have taken a tougher response from you, but you probably did the right thing. As for the wet carpet, would you prefer tenants who said nothing until the mold got established? NEVER, EVER, let a water problem sit. You know that.
Once you finish asking yourself those questions, then you can decide whether the tenants are unreasonable or not. 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.). Nobody moves in until everything is checked out and disclosed to the prospective tenant.
I had a landlord who was pissed at me because everything started breaking/leaking while I was there. His own repairmen told him that things had gone beyond their expected life cycles, and there was no sign of abuse. He and I stayed friends. Sounds like your air conditioner was just bad luck. I don't think I would have given the tenants a portable conditioner, but you did the right thing, and if the tenants raise cain later on something, it's an example of you trying to do the right thing.