It is the BE junction diode of T1 that does the rectification. So the Base of T1 does carry both RF and LF current. Because of you need LF current in the base also, the coupling capacitor C2 must be large to allow LF current. If you would change this capacitor to a value that only passes the RF current, you will not have your LF current in the emitter (hence no good detection).
If you want to drive it from another RF AM source, you need a LF path to ground (that doesn't short circuit the RF signal of course).
You are correct, when you remove C3, it will no longer rectify efficiently. The parallel resistor is to provide a DC bias path for T1.
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If you want RF amplification, you need to put it in between T1 and the ferrite rod with capacitor tuning. You may need matching to avoid that the RF amplifier loads the Ferrite LC circuit too much (reducing sensitivity). What frequency will you use?