There is hot water in the water heater tank, but the hot water pipe going to the house is cold. I have no hot water.

Here is what should be going on.
There is a length of pipe between the hot water that is in your tank and the nozzle you are expecting to get it out of.
It is really possible the are nearly 3 gallons of water in those pipes, especially if the water heater is many feet from the faucet.
Now about the drop in water pressure.
Try the cold water.
Does it do the same as the hot? Does the pressure drop after 2 - 5 minutes?
If so, suspect your pipes are clogged with sediment. How old are the pipes and the house/apartment?
Does the water contain minerals that clog pipes?
If the cold water is good, and the hot not, I should suspect there is blockage in the hot water pipes, especially going in to or coming out of the water heater., or perhaps the faucet is clogged with rust or deposits.
God bless your efforts.

Could be. Location of the pipes has to start from the farthest part of the house to the place where the heater is at.

If tank is rusted out, then that is caused by depleted anode rod, or tank that is not electrically grounded on hot side pipe.
If pipe nipple is rusted out, then that can be caused by improper connection between copper and galvanized pipes, and caused by non-grounded hot water pipes.

If tank is leaking, replace it.
If nipple is leaking, remove with pipe wrench, apply teflon tape or joint compound on threads, and install new nipple. Replace anode rod if possible. Correct copper-galvanized connections. Add ground wire between hot and cold pipes.

http://waterheatertimer.org/Leaking-water-heater.html
http://waterheatertimer.org/Replace-anode-rod.html

1) Look at the top of the old tank.
There will be markings on the tank, probably embossed in the steel cover that show cold - hot.

2) Many tanks have red and blue plastic pieces surrounding the inlet pipes

3) If there is a shut off valve located above tank, it should be on cold water line assuming house was plumbed to code.

4) When we install water heaters in old houses where water heater is removed, we look for shut off valve above tank that tells us which line is cold. If tank is missing and pipes have no shut off valve, we turn on outdoor water supply and see which of the two water heater pipes starts running water first ... because that is cold side.

5) For do-it-yourself water heater information about electric water heaters:
http://waterheatertimer.org/DIY.html

Add a comment for more free help.
Also take advantage of fixya phone service.
For a price, expert speaks with you over phone while you work on water heater or any do-it-yourself project.
Fixya is always less expensive than a service call.

You have a clog in the hot water line.
If problem is at one faucet, then remove aerator and clean, then disassemble lines underneath sink.

If problem is at every faucet, then that moves search close to water heater.
Look at hot water line that exits water heater and trace that line to first Tee.
Look at cold water line that enters water heater and trace that line to Tee.
Clog is located between those 2 Tees.

Buy pressure tester that connects to hose thread.

Connect pressure gauge to hot water heater drain valve.
Then conduct same test you did at faucet.
See if pressure drops at tank.

If pressure drops, then clog is on cold side of tank, and probably hot side too.
If pressure does not drop, then clog is on hot side of tank.

Suspect 1 is any place two different metals join. For example galvanized pipe and copper.
Suspect 2 is supply pipes that exit top of tank. Take apart and look inside pipes for corrosion and sediment lining pipe walls.
Suspect 3 is elbow. Disassemble pipes and look for sediment and flakes of rust.
Suspect 4 is all suspects above, each having sediment and corrosion.

Add a comment for more free help.
Also take advantage of fixya phone service.
For a price, expert speaks with you over phone while you work on water heater or any do-it-yourself project.
Fixya is always less expensive than a service call.

You have clog in the water line.
Since all hot water pipes are involved, clog is probably near water heater.

Buy pressure gauge at hardware store that threads onto hose connection.
Brady BTG 100 at Amazon
Picture of gauge

Check outdoor pressure for baseline pressure in your home.
Turn off electricity to hot water heater for an hour or two so there is not scalding water.
Open drain valve on water heater to see if water heater is clogged with sediment.
http://waterheatertimer.org/Water-heater-will-not-drain.html

Put pressure gauge on drain valve and compare with outside pressure.
Open hot water spigot on bathtub and see if tank pressure drops ... if tank pressure drops significantly with bathtub spigot on, then clog is on incoming cold water before tank, and could be on outgoing hot water pipes too.
If tank pressure stays about the same, then clog is on outgoing hot water pipes.

Clog can be sediment caught in cold water shut off valve located above tank. Open and close valve and test pressure again. Take apart incoming cold water pipes above tank and check for sediment. Replace valve. Look at condition of cold water nipple on top of tank.

Sediment caught at hot water elbow near tank.
Sediment can fill up old pipes. Take apart hot water pipes above water heater and see condition.
Look at condition of hot water nipple on top of tank.
You may have to take apart pipes up to first or second elbow.

Pipes can decay at joint where copper pipe connects to galvanized pipe. Check for copper to galvanized connections on cold or hot pipes and replace section of galvanized pipe. Copper pipe will probably be good, the galvanized pipe will deteriorate. Replace copper-to-galvanized joint with 3/4" non-conducting union from hardware store

It sounds like two problems.
Problem 1) Clog inside faucet or clog inside pipes at faucet
Remove supply tubes leading to faucet and see if problem is inside pipes or inside faucet.
My bet is clog inside faucet.

If you have galvanized pipe that meets copper pipe, this can be trouble spot.
Article on clogged water pipes at Inspectapedia:
http://www.inspectapedia.com/water/Water_Pipe_Clog_Repair_Guide.htm

Article on clogged pipes:
http://www.waterheaterrescue.com/pages/WHRpages/English/Troubleshooting/Tanklets/backflushing.html

Problem 2) Water heater leak ... you say tank valve is leaking? Is that cold-water shut-off valve? Or tank drain valve? Or TP valve?

If cold water shut-off is leaking, then replace shut off.

If tank drain valve is leaking, then check if plastic valve is cracked. If drain valve is cracked, then replace with a brass valve from Home Depot. If plastic valve breaks off, then use hammer and screwdriver to gently chip out the broken plastic valve. New valve threads need teflon tape to seal pipe. If valve is just dripping, then put garden-hose cap over end of valve and screw down tight.

If TP valve is leaking, then replace with same temperature and pressure rating. New TP valve threads need teflon tape to seal pipe.

Problem 3) If pressure is low all over house:
Buy pressure gauge at hardware store that screws onto hose connection.

Test pressure on outdoor spigot.
Open faucet and check if pressure drops. This says if problem is inside pipes that enter house.

Test pressure on water heater drain valve.
This says if pressure problem is before -or- after water heater.

If problem is before the water heater, then shut-off valve located on cold water line is suspect.
If problem is after water heater, it could be in hot water outflow pipe on top of water heater, or a nearby elbow.
Remove hot water line leading from water heater and check pressure. This says if problem is where hot water leaves tank.
Read articles above about clogs inside water lines.

Tanks are marked with Hot and Cold.
1) If water connections are on top of tank, look for red marking or word HOT that shows where hot water pipe connects. Look for blue marking or word COLD that shows cold water connection.
2) If water connections are on side of tank, cold water connects to bottom one, and hot connects to top.

3) If hot and cold are reversed, the tank will not work.
Pipes on top of tank: cold water enters tank and is sent to bottom of tank in a long tube called the dip tube. This means the cold is on the bottom since hot water rises to the top. The hot water exits top of tank.
Pipes on side of tank: You want cold water to enter bottom, so hot water naturally rises to the top outlet.

That is normal, the heat just rises from the hot water, the pipe on the right should be cold and if you were to run hot water anywhere in the house it will immediately cool of the cold inlet pipe and you can feel it