It shouldn't cause issues, since the water will remain in the heater without any pressure to push it out, there's no difference between leaving it sitting with the valve open or closed.

Now, draining the tank could actually cause problems.... But there's not any reason for you to do that for what you're trying to accomplish.

Turn off the power to the heater before doing any work

As long as there's water inside the heater it's fine.

Okay, then yeah, it’s pretty typical to just shut off the cold water feed to the water heater. Pull the cord/flip the breaker, whatever it takes to turn it off… open all the taps if there’s water pipes above it so everything drains, then replace the faulty valve in the tub.

Same concerns though, if the furnace isn’t properly vented you’re going to be breathing carbon monoxide. That absolutely must be corrected, the unit is not safe to occupy and any municipality will red-tag the installation and shut off the gas supply until it’s corrected.

Welcome to Tennessee construction. I never thought I'd think Florida was decent

(Obviously a joke, one bad worker isn't the sum of the whole)