Viability of Voltage Controlled Amplifier circuit

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.

First, you mix voltage and current. Voltage is the "pressure" on electrical charges, so they want to move. Current is the flow of electrical charges, if the find a way.

Case 1:

To answer the question, have a look into this datasheet of a common BC337

The table on page 2 states a Collector Cutoff Current of 100nA for V_CE=45V and V_BE=0V.

So yes, there will be a current through the collector. Keep in mind that a NPN transistor looks like two diodes connected back to back:

^{simulate this circuit – Schematic created using CircuitLab}

While this doesn't describe a transistor in general, here it does. You are operating the upper diode in reverse direction. In this case, real diodes will show a small leakage current, 100nA is a perfect sample for it.

You didn't say where E should be connected to, but if it's connected to GND, as in my schematic, the lower diode is shorted by the connection between B and GND, so the 100nA will flow through the base leg. It's just the same as if E is not connected anywhere. If E is also connected to Vcc, you just get a second diode in parallel to the first, so about twice the current (depending of the characteristics of the diodes)

Case 2

Well, if only B is connected to Vcc, and the rest is not connected to anything, no current will flow. If you connect E and/or C to GND, the diodes will be operated in forward direction, and a significant current will flow (and destroy the transistor if currents are niot limited by resistors)