Hot water heater overflow leak

It’s obviously leaking. If Liner is leaking replace heater. If piping is leaking, replace leaking pipe.

Yes you have a leak on hot pipe or TP overflow tube.
Put ear against pipe to listen for sound of running water.
http://waterheatertimer.org/How-to-troubleshoot-electric-problems-with-water-heater.html#listen
If TP valve goes into drain line, check if water is running down TP overflow tube.
Check each faucet for leaks.
If pipes are under slab, put ear against each faucet to determine which faucet is closest to broken pipe.

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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

No.
Water should not leak out of water heater.
Water should only come out of the TP valve and down the overflow pipe if water is too hot, or if TP valve is bad, or if you have stacking condition.
http://waterheatertimer.org/How-to-replace-TP-valve.html
http://waterheatertimer.org/Stacking.html
http://waterheatertimer.org/Leaking-water-heater.html

Problem is not clear:
Sounds like you have two different problems, but overheated water heater can cause both problems:

Before calling service technician, open following links for full do-it-yourself troubleshoot.
ADD A COMMENT for more details after looking at information below.
1) No hot water could be bad thermostat, or bad element, or circuit breaker is tripped.
Open following link to troubleshoot electrical problem with electric water heater:
http://waterheatertimer.org/How-to-troubleshoot-electric-problems-with-water-heater.html

2) Leaking water heater.
Open following link to identify problem with illustrations and troubleshoot leaking water heater:
http://waterheatertimer.org/Leaking-water-heater.html
Look carefully at illustrations on page:
If drain valve is leaking, then put a garden hose cap on it.
If TP overflow pipe is leaking, do NOT cap that pipe

Hot water pressure relief valves serve two functions. The first is to remove high-pressure water from the hot water tank. The second function is to act as a high temperature valve to remove water that exceeds a predetermined point for the valve. Both of these relief settings are preset at the valve factory. Many pressure relief valves will come with a metal tag that identifies these release points. If you're attempting to remove the hot water heater pressure relief valve, pay careful attention to ensure that you replace with the correct valve.

Your hot water heater has a pressure relief valve safety mechanism. When pressure inside the water tank exceeds certain limitations, the valve opens up to allow the pressure to be released. A faulty pressure relief valve will not release excess pressure, which could cause extensive damage to your home. You should test the hot water heater pressure relief valve once every six months to ensure proper operation. If the pressure relief valve fails the test, replace the valve with a new one immediately

Locate the pressure relief valve on your hot water heater. The pressure relief valve has a lever handle on top with a drain pipe attached to it. The pressure relief valve is most often on the side of the water tank toward the top. Newer models of hot water heaters have the pressure relief valve on top of the water tank.

Inspect the area around the pressure relief valve for signs of leaking water. If there is leaking water near the relief valve connection, replace the relief valve. If no leaking is evident, proceed with testing the pressure valve.

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Examine the area below the pressure relief valve drain pipe. Most water heater closets either have a drain in the floor below the pipe or the pipe is inserted into another pipe to direct the water out of the home. If the end of your drain pipe is not in another pipe or there is no drain in the floor, place a bucket under the drain pipe to catch water when you test the pressure relief valve.

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Pull the lever handle on the top of the pressure relief valve up with your fingers. A working lever will lift up without difficulty.You should begin to hear water going through the drain pipe. Slowly release the lever to shut off the pressure relief valve. The pressure relief valve is good. If you did not hear water running through the drain pipe, lower the lever on the pressure relief valve and replace the valve. If the lever is difficult to raise, do not force the lever. Replace the pressure relief valve.

Now with all that said if you have a closed loop water supply system and you have a pressure vessal/expantion tank it may have gone bad and is making the TP valve leak. you can call your water provider to see if you have closed loop system.

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.

This is not an "overflow". It is a Temperature/pressure relief valve. Sometimes they leak with age or build up solids in valve preventing them from closing completely. This is the single most important safety device on your water heater. If one of the thermostats is malfunctioning, the TPR valve will relieve. If the pressure rises too high, it will also relieve. You can replace the valve, but be sure it is the same temperature and pressure rating. You will need to turn the water off and drain some of the water prior to removing the valve. Don't forget to reinstall the down pipe that directs floor to floor or drain. If a new one leaks also, suspect your temperature control. Water heaters can be bombs if the safeties are not working properly

sounds like your pressure relief is leaking. Make sure your home water pressure isn't abnormally high (above 80 psi). As long as your water pressure is fine, you should replace the pressure relief.

remove and, replace the pressure/temp valve. This valve is for safety and, if it is leaking it must be replaced immediately.
It has a small lever on it and, when you lift it water will flow out, it is a right angle brass valve located in the side of the water heater.