Sometimes it will back up from a dishwasher (machine)
Replace the faucet.
run it under hot water in the sink
It sounds like the water heater is so far from the kitchen sink that a recirculating line may be required to keep the line feeding the sink hot, otherwise, it will cool down causing you to have to run all that cool water out before the hot gets there. The reason you get it faster when you run the bathroom is because you are drawing more GPM through the line with two fixtures on.
When water disappears down a sink, it goes through pipes and either goes into a septic system in the city or a septic tank in the country.
Neither, because you want to lay down on your back and it is impossable to sink in quicksand. But from walking or running your would walk.
Probably not, depends on where the clog is at and how the dishwasher drain is connected. I wouldn't chance it, but the worst that will happen is the sink will fill with the water from the dishwasher.
If indeed it is IN your sink, simply run lots of hot soapy water through it.
You probably have a blocked or frozen pipe into the H/W tank.
The discharge line (soil ) may be slightly back pitched so it takes longer to drain
I assume it is the bathroom sink. Is there hot water in another sink in the house? In old steel pipes I have run across ones that get so rusted on the inside that water can not get through them. If anything has been done to the hot water line anywhere in the house, sediment or rust may have broken loose and plugged the fixtures.
major back up in the lower end of the house (meaning that the kitchen sink sits higher than the bath tub that was getting the overflow) meant that everything that was washed down the sink would come up through the bath tub drain in about 30 secs of having kitchen sink water running. bought some industrial strength crystals for drainage blocks and poured it in both drains. let the sink run with hot water and watching about 1/4 of the tub fill up with years of gunk! after we stopped the water the bath tub emptied and we haven't had a problem since!
Without the water cycle we would eventually run out of water. The water cycle is a very important process.