This was a new one for me. The shower valve in the hall bath was only hot first then would not adjust at all and only mild water came out.
I would expect the cold water side to be less than 80 degrees, correct? 
Not sure what shower you’re talking about, but the WC is probably on a Tempering (or mixing) valve, and may be defective.
I ran across this once and what happened was just the opposite. The hot water got really hot then luke warm, then intermittently got hot then warm. What happened was that there was a hot and cold line crossed somewhere in the concrete slab. The entire house had to be replumbed because they could not access the water lines in the concrete slab.
This one is an older home with a cellar/basement. Most lines are visible but the hall shower valve was defective. I suspect thats the culprit but let the plumber figure it out. 80 deg water can be confusing. You think its getting warm but never does.
I found this on a few occasions in homes. Typically a Hot water recirculating system installed on a single story home. Timer pump at the water heater and a temp actuated bypass valve located at the fixture farthest away from the water heater…
The system is described well here: http://www.askthebuilder.com/413_Hot_Water_Recirculating_Pumps.shtml
Hence the moderately warm water at the toilet!
But lack of hot water at the shower may indeed lead to some other issues present especially of this type of system was attempted to be installed by a home owner.
This was someones polished turd (AKA flip) the home was built before the hippie movement so no recirculation pump on this one.
If this bathroom was close to the water heater, it could be pushing warm water back if the water heater is turned up all the way. Only of course if the water heater does not have an expansion tank or tee in the line.
Don’t know about the shower, but is it possible the toilet was mixed somewhere to prevent sweating? Was this on a well?
Or it could just be fubar …
Tim