In reverse order:
Fuel or Lox cooling may take place in various locations, depending of the engine design. It could be the bell, or it could be the head of the combustion chamber. Wherever heat will be the biggest problem.
Lox (or whichever liquid is used for cooling) doesn't boil off due to the pressure in the system. Not unlike the water in pressurized water nuke plants (when they work right) - the water can reach 2-3x normal boiling temperature and not boil, so long as it is highly pressurized. Or more simply, the way a pressure cooker can operate at above-normal boiling temperature (to cook food faster).
The thrust chamber is basically the whole "obvious" part of a rocket engine - the combustion chamber where the fuel ignites, AND the flared exhaust nozzle (bell). Any part of the engine where combustion creates enough overpressure to provide thrust, or in other words, any part of the engine that has to withstand the massive thrust forces and transmit them to the vehicle without coming apart.
The turbopump is simply a fuel pump that delivers a controllable and precise amount of fuel and fuel pressure (and oxidizer and oxidizer pressure) to the combustion chamber.
While the fuel/oxidizer tanks are pressurized, and fuel will also flow backwards/downwards under acceleration or gravity, their ability to deliver the right amount of fuel/oxidizer is too random for controlled flight. As fuel or oxidizer get used up (tanks get emptier), their "head pressure" will drop - and as fuel weight decreases, the acceleration (and thus the flow of fuel under G forces) will increase.
The turbo pumps eliminate these variables, and deliver a constant "engineered" amount and pressure of fuel/oxidizer, regardless of tank pressure or other influences.
There are - in the J2 - control valves downstream of the turbopumps to provide even more precise "throttling" of the engine output.