Transistors – Fluctuating Frequency When Using NPN Transistor

You can use the Shockley equation, which is another idealization, but a better one than a fixed diode drop. To apply it, you have to know, for the given diode, all the parameters:

$$I=I_\mathrm{S}\left( e^{V_\mathrm{D}/(n V_\mathrm{T})}-1 \right)$$

I would suggest to replace the transistor with an actual diode model, from which you can pull out the parameters and calculate by hand.

In this answer to a question, I was working with the formula, in relation to the behavior of a diode model in CircuitLab. See the question's "appendix" at the bottom.

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.