Is a tank-style hot water heater always under pressure?
In other words, is the entire tank pressurized?
Is the pressure largely created by the force of the incoming cold water supply?
Yes, the tank is under pressure from the incoming cold water supply. After all, the incoming cold water is what is pushing hot water out of the tank and into your faucet.
However, there are usually two safety valves:
Temperature/pressure relief valve that will let the hot water or steam out, if internal pressure or temperature goes over a certain limit.
A vacuum breaker valve that will let air in if internal pressure is less than external (air) pressure: prevents the tank from collapsing if there is no cold water coming in and you're pumping hot water out.
Yes, the water tank will be equalized to the same pressure as the water in all your house plumbing, unless there are regulators or check valves to interfere with that. And this will be equal to municipal supply with the same exceptions.
If water is flowing, it gets a lot more complicated.
If you shut off main supply to your house, it will remain at that pressure until you crack a valve open and let off the pressure, or leaks have the same effect. A house can hold pressure for months if it's tight. A tight house with a leaky supply valve can repressurize even after you thought you bled off the pressure. It can also fill pipes you are trying to solder LOL.
The only exception is if there's a regulator or check valve which prevents equalization. In this case, your tank and household plumbing can actually overpressure above supply, typically because you have a load of cold water in your water heater, and it heats up and thus expands. This has no limit as to pressure, and can burst piping.
Since everything is equalized, the safety valve on the water heater protects the whole house's plumbing, as does a vacuum break if you have one. Vacuum is not as dangerous to piping because it cannot be more than 15 psi or 1 atm. You would think hot water heaters would be vulnerable to vacuum, but home biodiesel brewers use them as vacuum dryers.
If the main water valve, and any other valves leading to the heater are open, then the pressure inside the tank will be equal to the supply pressure. Once the tank starts heating, however, the pressure will increase.
If you have an old system that's never been updated, you'll possibly not have a check valve on the main. If this is the case, the extra pressure will be able to push back into the supply.
If there is a check valve, you have a closed system, and there's nowhere for the extra pressure to go. In these situations, an expansion tank is required. The expansion tank will absorb the extra pressure, and reduce stress on the system. However, the expansion tank is not infinitely large, and so can only absorb a limited amount of excess pressure.
If the pressure builds beyond what the expansion tank can handle, the excess pressure will begin to build. At some point the temperature and/or pressure in the tank will be greater than the set point of the temperature and pressure (T&P) valve, at which point the valve should open.
Once the temperature and pressure drop back down below the set point, the T&P valve should close. Though sometimes once they open, they get stuck open. In this case, the valve will need to be replaced.
If your t&p pops off then you have an issue needing addressed. It really shouldn't happen very often, meaning very rarely. If there is an expansion tank then it's bad. The bladder in the tank has ruptured and it needs replaced. Flick the expansion tank with your finger, not hollow, new tank and t&p.