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Obstruction in the hot water supply line, bad valve, or broken water line. Drain system and remove valve cartridge, open hot supply to blow out line and see if water comes out. If it does than you have a bad cartridge, if not broken line that needs to be repaired.
Iron? == == Possibly a slight blockage allowing toilet water to ernter the waste line of the tub
Repair it or line it with a plastic bathtub liner.
YES, you do need a trap. Usually the bathtub drain is on the same sewer line as the nearby toilet.The smell from the toilet's discharge will sneak up through the bathtub's drain line without a trap ... not pleasant!.
remove alt and ac belts,then remove the black cover behind them make sure you line up the marks on the pulleys and castings when reinstalling
Yes and a vent and hot and cold water to it and a waste line would also be a great idea
To remove and/or prevent "Water Hammering" in your water lines.
yes
Install a checkvalve
The sewer line is plugged or collasped. The sink and bathtub may drain because they drain at a slower rate. Toilet is all at once and the drain can't handle it. Water has to go somewhere so it backs up in the sink and tub.
it is run off of a different line from the main water supply. you may be able to trace it from the hot water tank and see where it branches off.
Remove the top water line from the radiator to the water pump. The thermostat is found there.