Not being able to see the pipe and not knowing where it is, it's kind of hard to know. If it is a cold water line and you have not shut off the hot water tank, the pressure from the hot water will back flow and hot water will come out of the cold side.
Hot water pipe rust first as compared to the cold water pipe for the same chemical composition of the water.
Cold water pipe.
I would use hot water conn. on it. Both on cold and hot
This is to allow for expansion of the hot water-- otherwise the cylinder would blow up.
Cold water pipes would be the sweaty ones.
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.
If shut off the hot water valve then you take off your supply line from your hot water valve to your faucet and then turn your hot water valve on slowly (with of course a bucket or something to collect the water under it) and if you can see you have water pressure and flow then most likely the problem would be in your faucet on the hot side. If you have no pressure or flow when you turn the hot valve on then it could be debris plugging your hot water pipe or your hot water service valve (angle stop, straight stop) which older galvanized pipes are known to do so. To fix the problem you would need to find out where the problem is then either fix or replace the faucet or replace the angle/straight stop or try back flushing the hot water pipe and or replace the pipe with pex or copper pipe.
air lock in hot water pipe
Vent, return circulation
Check pipe size for hot water or if galvanized pipe used, pipe could be clogged inside with rust and scale.
There is no special PVC pipe. CPVC can be used for hot water applications or in my opinion just use PEX pipe.
The water is probably frozen in the pipe