Just purchased with signs of rusty residue on the engine...After driving it home, there were also rusty water beads on the front end exterior...

Coolant and oil levels seems good - don't see any pools on the ground.

I switched the radiator cap, and it was driving fine (for about 2 weeks), then recently started overheating after about 5 mins of driving...

Does the rust indicate some kind of vapor leak when the engine heats up?

 

Tell me you flushed all the rust out of it? If you haven't then flush it, replace the thermostat, and check all hoses for leaks and make sure the belt is tight.

 

There are a lot of reasons for this. So, babysteps....flush the system out, the previos owner may have not changed the coolant since the factory! also, be sure to use distilled water as it does not containany of the minearls and other crap in the water which attacks metal.
Also, be sure to use only Mazda thermostats and radiator caps.
-a

 

Any idea why the overheating problem appeared so abruptly and recently?

 

The change in cap could have fried the thermostat if it's not the right one, which would cause overheating. The rust in the water is all the **** in the core and irons that has built up. I might even pull the radiator and get it cleaned professionally just to do the job right. Then I'd flush the crap out of the system (pour distilled water through the upper hose and let it flow through the engine untill it comes out clear, apply pressure if needed). I'd then replace the cap with an OE one as well as an OE t-stat and a new gasket. Just for the sake of doing it right and once, you may as well replace all the hoses and clamps since you just bought the car and piece of mind is cheap.

EDIT

Rusty water on the exterior can mean that the overflow bottles cap popped off and spewed water out, but you'd probably know if that happened, it makes a good bang and the engine would have been way hot.