how deep do these submarines go
All the way to the bottom.
The models are usually run at depths of 3-5 feet (1-1.5 meters) in most cases. The chlorine and other chemicals in swimming pools reflect the radio waves at depths of 3-5 feet; below that depth, the subsafe (failsafe device) - if installed - will blow the ballast tank and send the boat to the surface.
how do they stay under there
If the question is "how long
do they stay under; the answer is a long time, much longer than the batteries will last if you have a major malfunction.
Operating time depends on the Ah (or MAh) rating of the battery, generally speaking 100 MAh = 1 minute of operating time.
If the question is really "how do they dive" the answer varies. Dynamic diving models rely on forward speed and the dive planes to dive. Dynamic divers have a minimum hull speed, slow down below that speed and the sub will slowy surface. Static diving subs have a working ballast tank and use several different methods to flodd and empty the tank. Reversible pumps are used in some models, with the periscope acting as an air vent. Some boats use compressed gas (usually air brush propellant) to blow the ballast water out of the tank. These gas boats have a separate vent at the top of the ballast tank that opens to let water in. Some boats use a piston assembly, much like a hypodermic syringe. Other boats use a blood pressure cuff inside the ballast tank to displace the water.