The pipes are not empty when you shut off the water, there just isn't pressure to push the water out the faucets. Per your comment above, there are two floors of pipes and fixtures above the heater. They remain full of water when you turn the supply off.
If you are cutting pipes to work on the tank or replace it, and it has shut offs of its own on both sides, you can turn them both off and all the water remaining in the pipes above will be held back. If the heater only has one shut off even if it's the cold one, you should turn off the main supply to the house. You are going to have to drain the lines above the tank in order to work on them. To drain them you have to open faucets and if the main supply is still on, the faucets will feed water back into the tank through the hot pipes.
Turn off the main cold supply to the house, and turn off the tank heater at its switch or knob. Then, empty all the pipes above the tank before you cut them so water doesn't pour on the tank, possibly wetting the electronics or insulation. To do that open hot and cold faucets on each floor and use the tank drain with a garden hose.
There's no way to know when just the pipes are empty. The tank will gush out through the hose but the pipes will empty relatively slowly. So empty the whole tank. There are other ways but they are more work.
Recap: Turn the main off, turn the heater off, empty the tank, then cut the pipes.
Your question about using a mixer valve as a shut-off seems irrelevant to me, and I think the points in this answer hopefully show you it's not a good idea even if it works.
When your work is done, turn the water back on, and run hot and cold faucets on each floor until water and no air comes out everywhere. Then restart or relight the heater.
