At the showing, someone used one of the second floor restrooms and the toilet malfunctioned when refilling, with the tank overflowing onto the floor.

If not and you have a pump somewhere in the system it could be malfunctioning and that would be a reason for the frequent pumps.

For example, my lease requires under the "Maintenance" section, among other things, that the "Tenant shall immediately notify Landlord, in writing, of any problem, malfunction or damage." 

(Notice of any malfunction must be reported within 5 days of occupancy.

This does not include regular wear and tear or issues caused by malfunctions of course.Accidents however happens and if its a good tenant and we have no other issues with them, i might just pay the bill and have it as expense.

The lease should also list who is responsible for maintaining the equipment in the event of malfunction, i.e. automobile lifts breaking down.

Is it possible it was a malfunction of the thermostat/heating system or did the upstairs tenant really turn the temperature down so low that the pipes froze?

That will get them moving.Also, background and credit check, security deposit, Work information, previous residence information, Fix the house once, after that, "you break you fix" or "reasonable repairs of malfunction" (like roof leak or plumbing issues), have them sign it-- take pictures of the house before then tenant moves in, and take out of security deposit, based on the pictures you get when they are leaving --  you were too nice.   

The Seller did sign and give us a Residential Property Condition Disclosure where it was stated that the Seller was not aware of an defects/malfunctions in the central heating.

Your example of "what if I said the house was yours to keep" doesn't hold water because no one would consider that a fair exchange but many people would see the exchange of a month of rent for a couple weeks of a malfunctioning fridge reasonable, especially since you made the offer.