Electronic – Finding “beta” in an NPN transistor

Bipolar junction transistors (BJT) have 3 primary modes of operation:

1. Off: no significant current flows from the collector to the emitter
2. Saturated: the transistor behaves like a diode (which is why there's a fixed voltage drop across the collector/emitter pair under saturation)
3. Linear: this is where the transistor behaves like a current amplifier. The current flowing from the collector to the emitter is proportional to the current flowing from the base to the emitter. Note that in this region the voltage drop across the collector/emitter pair is NOT roughly constant, but linear.

In cases 2 and 3 there is current flowing from the collector to the emitter.

You can saturate the BJT by attaching a smaller load (i.e. higher resistance) than the linear limit allows. Think of a pipe with a valve. If there's something clogging the pipe upstream, at a certain point no matter how much you open the valve you're not going to get any more fluid flowing through the valve.

Yes, the electrons do literally move to generate the current, though this isn't directly used for most electrical engineering analysis.

For most EE analysis we don't consider the motions of charged particles directly, instead relying completely on the abstract concept of current. This is an approximation of reality, but it turns out to be remarkably good in most cases.

Ironically, in my day-to-day work with plasma physics we do consider the motion of charged particles instead of relying on abstract currents because the analysis does differ depending on what species are moving and where/how fast they are moving.

Rule 1 isn't a "good idea", it isn't a "guideline", it is a fundamental tenet of transistor physics. If for any reason (during normal usage) it is unable to hold then the circuit will not operate.

As for the lamp, it is a purely resistive element. It should have 10V across it, but thanks to the transistor it won't. So the transistor gets 0.2V and the lamp gets 9.8V and reality is saved.