Is it necessary to shut off the power to adjust a thermostat on an electric water heater? What happens if I don't? I’m asking because I don’t have access to my circuit breaker panel in my apartment complex.

It depends on the water heater. For most electric ones, it's smart to disconnect power before making adjustments.

Also, if the electric circuit breaker disconnect is not 'within sight' of the water heater, then there must be a disconnect switch installed near the appliance. Not doing so is a violation of the National Electric Code. You may want to bring this to your landlord's attention.

And finally, there is a provision in the Code that stipulates that occupants of a multi-family dwelling must have access to their electrical panel at all times unless the owner has a qualified person on site that is able to address tripped breaker issues at any time.

I would advise against adding to the existing circuit. First of all, it’s a violation of the Code. Secondly, you’ll be constantly tripping the receptacle’s circuit breaker due to overload whenever you plug in anything that uses a good deal of power (such as a hair dryer, etc.).

Also, heating circuits are considered continuously operating loads; thus, they must be installed on circuits rated for 125% of the maximum wattage. Most bath receptacle circuits can't handle this load.

Do yourself a favor and play it safe: install a GFCI dedicated circuit to handle the heating load.

Is it possible to control two different baseboard heaters on two different circuits with just one thermostat?

Yes. You can do this with either a heating relay or a multi-pole contactor of some sort. In either case, the thermostat controls the coil of the relay (or contactor). When it is energized, the magnetic coil then pulls in the contacts of the heavier load (the actual baseboard heaters).

Yes. Wall-mounted electric baseboard thermostats with an OFF position are quite common. You can usually find them at your local hardware store or home improvement center.