Let us look at a car engine; maybe this will clear things up a bit.
Such a piston engine cannot start to run on its own; it needs to have its crankshaft turned by some external power source in order to get the air/fuel mixture, ignition, oil system etc. in motion. As soon as there are the first one or two charges igniting in the cylinders and driving the pistons, the engine can run on its own and no assistance from outside is needed.
The external power source used for this is typically an electric motor that will turn the engine; its power normally comes from an accumulator.
In analogy to this: Neither can a jet engine get running on its own. Its shafts will need to be spun up to a certain speed by an external power source so the compressors feed the combustion chambers enough air. As soon as there is a steady flame in the combustion chambers (requiring a constant supply of air and fuel), the compressors are kept running by the power delivered by the turbines and no outside assistance is needed any more.
And also here: You need some means to get the compressors up to speed. An electric motor is definitely eligible but comes at some costs that makes it less than optimal in most cases. So a pneumatic motor is used instead. The place of the batteries is taken by the APU here.
As soon as the starter has done its job of accelerating the engine (be it piston or jet engine) to a self-sustaining speed, it is disengaged. The power needed to sustain the engine then comes from the power stroke (in case of a piston engine) or from the turbine section of the jet engine.
I would advise You to get a good understanding of gas turbine basics first. Many postings in this thread contain links that will help You a lot. And once You have a good grip of what happens in a steadily running gas turbine, it will be the time to look at how a gas turbine is started, governed and operated. For Your question 3, You will likely find
this thread of interest.
EDIT: I found a cutaway picture of a pneumatic starter on
this page.
The APU bleed air enters from the left, passes a row of stator vanes before working on the turbine and is then vented via the annular grille overboard. As such a turbine runs at rather high RPM, its speed is translated to a lower RPM and higher torque (remember, p=T*n*const.) in order to drive the engine shaft: this is why there is a planetary gearbox installed. Typically, such a starter also has a clutch that allows the starter to drive the engine but keeps the engine from driving the starter.