What does a wastegate do? And what about the different sizes?

 

a turbo is powered by exhaust gases, so as it pushes more air into the engine, there is more exhaust coming out, right? so the turbo will keep spinning faster and faster, meaning that, left unchecked, it will keep making boost until it hits its max air flow. a wastegate will open when boost gets to a predetermined level and it will bleed off the excess exhaust to keep the turbo from overboosting. the sizes are for different applications, a bigger turbo running higher boost will need a bigger wastegate to be sure it wont bottleneck the exhaust going through it. if it doesnt flow fast enough to bleed off all the exhaust, the boost will spike, and you can damage your motor.

 

Is there any way you can rig a wastegate up so that you can open it from inside the car, thereby being able to turn your turbo on and off?

 

not really. you could use a boost controller to turn the boost down, but then you start running in to tuning problems. and also, you can learn to drive it while staying out of boost.