1. Pick a stable oscillator that absolutely minimizes all capacitance other than the intrinsic C of the antenna (this includes the self-capacitance of the tank inductor).
2. A plate antenna can almost double the absolute sensitivity, and (depending on how you do it) can increase linearity.
3. ~50Hz bandwidth (after filtering / averaging) is a good target.
I spent a long time looking into period measurement, but ultimately abandoned it. A big problem with it is the measurement periods are asynchronous to everything else. Study the CIC filter closely, it is the equivalent of one or more integrate and dump filters (FIR filter where all coefficients are 1).