Hot water is leaking from under sink hot water turn on/off. have hot water heater, how can I fix?

This important part is right at the end, you have a well!! It sounds like you may have run it dry? Check the well and the pump first. The rest is stop valves and pipes to the tank and beyond and not likely to be causing issues. Your sediment issues and problems with the anode is also due to using well water which is nearly always loaded with minerals..

There should not be any issues with the manner in which you have connected it.

The water inlet to the heater must be plugged.

1) If water flow stops when tap is off, then Sound of water running: could be factory-installed heat trap inserts or nipples: These things cause noise.
http://waterheatertimer.org/Low-hot-water-pressure.html#trap
http://waterheatertimer.org/9-ways-to-save-with-water-heater.html#heat-trap
http://waterheatertimer.org/How-to-troubleshoot-gas-water-heater.html

2) If sound of water continues when tap is off, then there is a broken pipe or leak somewhere in plumbing system.

If you need further help, I’m available over the phone at https://www.6ya.com/expert/gene_9f0ef4df2f9897e7

Since you have hot water at all the other fixtures, the problem lies in the sink, not the water heater. It could be that the sink is farther away from the water heater than the other fixtures. It would take longer to get hot water.

If pipes are leaking, then that can be repaired.
If tank is rusted out and water escaping from the tank, then turn off the water plus power to the heater for safety, and then replace water heater.
Rusted water heater can be welded for repair, but remember that hot water is under pressure and tank can rupture any time.
Safest course if to replace heater.

Open following link for leaking water heater checklist:
http://waterheatertimer.org/Leaking-water-heater.html

I wrote a page on leaking water heaters with resources and information:
Page lists things to check.
http://waterheatertimer.org/Water-at-bottom-of-water-heater.html

Also run hand over tank to find warm spot that says where leak is coming from
A rusted tank cannot be repaired.

If your tank is rusted out, it might burst open and flood the area.
Hot water inside tank is under pressure.
Turn off heat source to water heater.
Turn off water valve located above tank.
Open tub faucet on hot side and relieve pressure from tank.

You can limp along with a leaking water heater by turning it on when hot water is needed.
For example a shower uses 5-9 gallons hot water.
Turn gas water heater on for 15-20 minutes.
Turn electric on for 20-30 minutes.

To conserve resources with next heater:
http://waterheatertimer.org/9-ways-to-save-with-water-heater.html

It could be back flowing into the hot water heater tank which is not full because of the leak. Even though the valve is closed on the cold water entry side, there is probably no valve on the exit side. That is how it is returning to the tank.

basically the principle of the heater is water flows into a internal tank and is heated by an element. You open a hot tap and the mains cold water forces the hot out and replaces it inside the heater.As you have a flow problem i can only suggest you may have some kind of restriction or blockage in the cold water supply at the water heater or from the hot supply from the heater.If you live in an area where lime content is high then this can calcify when heated.If you can, isolate the water supply and electrical supply and check the pipework for blockage.Also investigate your hot tap as this may be the fault. Good Luck