You have air in your pipes and it needs to be flushed out.
That's just how long it takes to get the water that is sitting in the pipe out. Unless you have a circulating pump on the line so that there is always hot water at the faucet, it always takes a bit of time for the water to get hot. The hot water is in the tank, not the pipe connecting it to the faucet. There is nothing in the pipe to keep the water hot.
Usually from a hot water tank.
Burn your tongue and throat is about the only issue. Hot faucet water is no different than cold water you heat on the stove or drink in your coffee.
Yes
Turn the water off to the faucet. Remove the handle. The top of the faucet should unscrew to get to the ball valve. Replace the valve and gaskets then reassemble the faucet.
Check the temp on your hot water heater, you can turn it up.
You go to the bathroom and run on the hot water faucet.
If you have an instantaneous hot water heater, and your faucet doesn't flow enough water it won't turn on the water heater. Thus requiring another faucet to be turned on.
Unless there is a recirculating pump the hot water has to come from the hot water heater. Depending on how far away it is from the faucet, it can take 15 to 30 seconds for the hot water to get there. Since there is nothing in the pipe to keep the water hot, it cools down between uses.
Your hot water line needs more insulation probably. It sounds like your hot water is cooling before reaching that faucet. It may be that the one faucet is not able to turn all the way to hot until you adjust the faucet (usually in the handle) of a 10 year old or newer faucet.
Kitchen faucet? If so, the sprayer attaches to the underside of the faucet in the middle.
Hot water tank not working or a faulty faucet.