Usually not. If toilet has a mixing valve going to it so warm water fills toilet tank and toilet runs constantly, then enough warm water could leak out through toilet to possibly have an effect on water heater.
Depends on how bad it is leaking
Yes, coolant has to flow correctly for the heater to function.
Take shorter showers, or get a tankless, "on-demand" water heater that heats the water as you use it.
Probably the hot water heater is off, either it has broken down or the pilot light has goen out.
Water heater fan? I have never seen a "water heater" with a fan. Possible a boiler, hot water heating system. Assuming you are talking about a "water heater" in a house. Please give a little more information on where it is and what it is intended to do.
Because the flapper is stuck.
My bet would be the hot water heater cycling water through
Someone hooked up the hot water supply to the toilet instead of the cold water supply- or your water heater is back drawing hot water through to the toilet from a cold water inlet (supply)
check that the toilet stopper has been switched on and check that the strainer inside the toilet inlet is clear of debris
By taking shorter showers and not flushing the toilet
Erosion
Either the thermostat has been set too low, your hot water heater is not big enough, or your hot water heater has not been installed properly.