Agree with you on all.
but number 4)A BJT is a transconductance amplifier-
that is true only in C.E ,but not in C.C or C.B.
I stand by what I said. No matter what configuration you set up, you are adding external components to the transistor. This means that you are no longer working with just a device. You are instead working with a transistor circuit
. I aver that the transistor within the circuit will still act like a transconductance amplifier, whereby the voltage across its base-emitter will affect the collector current. This is analogous to a op-amp which can be made to do a multitude of electrical functions by adding external components while the op-amp itself still is functioning as a dual high gain voltage amplifier.
The transistor doesn't know how it's connected
That's right. It still functions the same way, but its input and output is modified by the external components..
For common emitter circuits, maybe, but not for emitter followers. In the output stage of a power supply or audio power amp, base current flows directly into the load, and can account for 5% to 10% of the load current in some applications. Semantically, that might not be covered by the phrase "contribute to the amplification function", but I think that characterizing base current as wasted is a misleading generalization.
For a CC circuit, base current does not contribute to the current amplification. That in itself is a "waste". If we cut out the collector, then all the base current would drive the load and we would not have any current amplification. Furthermore, it would load down the base network whose purpose is to provide the desired voltage. So the fact that the base current still goes through the load on a CC configuration does not mean that the circuit is operating optimally. In a perfect world, no base current would exist and the current would depend entirely on the base voltage across the load.