If your water hose has been in the sun the water will be warm briefly.
Jebus a lot of people have bad information.
Two issues at hand here.
Definitely the toilet water valve has began leaking. Need to replace it. Change your ball valve seals while you are at it. Might as well make the toilet work as good as new again. Rarely do you need to change the entire toilet unless you just want to.
The warm water issue has to do with your water heater being hot therefore having more pressure than the rest of the water system (you were not connected to city pressure and your water pump was turned off). Even if the water pump was ON there is a chance the pressure in the hot water tank is higher than the output pressure of the pump. You see where I am going with this.
It has nothing to do with your T&P relief valve. To check the T&P just crack it OPEN. Does water come out? Does it stop leaking when it CLOSES? If both of these answers are true then the valve is just fine. If either is not true then replace it.
Nothing is wrong. Fix your toilet and get back to business.
Sounds like your hot water tank over pressured your water system. Check the T&P valve on the heater. You may want to add an expansion tank.