They should have no effect on each other. They have separate circuits and separate fuses.
It is a water heater, not a hot water heater. There would be little point in using a hot water heater. I assume that is your point.
Bridgestone Dueler tires are made for tough road conditions. They could be used as winter tires so heavy snow or rain would be conditions they would perform well in.
I do not see how using the heater would effect MPG. The AC on the Prius is all electric, there is no compressor to put drag on the engine.
This is an indication that your heater core is leaking. To stop the leaking you will have to replace the heater core or stop using your heater.
Sounds like you have a leaky heater core. You will need to replace the heater core and give the cooling system a good flushing, it shouldn't be rusty.
I do not think you would require all terrain tires for city driving. This type of tire is usually sold for off road driving using trucks, jeeps or recreational vehicles. I think all season tires would suit your purposes.
New tires actually have less friction on dry surface than old, bald tires, but once the old tires hit water, you new tires quickly win.
first check that the tires are balanced also could be the design of the new tires
The purpose of a heater is to heat up the room when it is cold; by using gas
One benefit of using a radiant heater instead of a convection heater is that they can heat an entire room for a longer period of time without burning out the motor or shutting off.
The main disadvantages of using the cheapest tires available is the loss of grip under certain conditions like torrential rain as the water cannot clear as quick as more expensive tires. Another disadvantage is that cheap tires are normally remoulded.
You can find mud tires for reasonable prices online. I would suggest using the following website - http://www.justtires.com/tires/Mud-Tires.uts, however there are multiple options. I chose this one as it has a detailed search option and large variety.