As a Commercial HVAC Mechanic i've had plenty of experience with water heaters, solar collectors etc. With that said, I'll try and answer your questions;
Q:Will I ruin my electric hot water heater if I let the sun heat it?
A: Probably not, but the design for the outside of the heater (electrical components) are mostly rated for 125 Deg F max temp, if the temp is up to 140-160...shouldn't be a major issue. They might even be 175 Deg F, some are stamped if you want to check.
Q: From someone else;I think the thermostat for the hot water tank might prevent the heating element from turning on.
A: No problem there, the sensing bulb is in the tank itself just the switch is on the outside of the tank.
You didn't say if your electric water heater is a natural vent style, no powered fan for venting? If it is just a standard heater then you shouldn't have an issue. With the fan you would shorten the motor's life with that high temp in the area.
If you are going to do that, i would remove the outside shell of the heater and remove the insulation. Maybe even paint the tank black but this will expose all the wiring which will make if dangerous.
I personally would do a simple pre-heater, you don't necessarily need to run a circ pump if you don't want to, a basic pre heater would be cheap to construct...could even put it on the door and run some braided flex hoses at the door hinge. You run your cold water through that first then to your water heater, the more capacity you have the more efficient it will be.
If you started with a glass door you could build it on the backside (inside) of the door, simply box it it and paint it black. I plan on building a simple pre-heater using my old water tank as a storage tank, I just put in a tankless a few weeks back. I'm simply boxing in the tank, paint everything black and pipe it into my inlet for the water heater. Add a couple bypass and drain valves and your good to go.