1. The equivalent circuit of a crystal is a very large inductance with series and parallel capacitance. A crystal has both series and parallel resonant modes, usually close together in frequency, but some are optimized for parallel or series operation. Since the crystal is like a tuned circuit, you can put capacitance in parallel with a parallel resonant crystal and change its frequency (a little bit). Parallel resonant crystals always have a shunt capacitance specified that will put it at the specified frequency. You can make a Colpits oscillator using a crystal but it would be too dificult to tap the crystal to make a Hartley oscillator.
2. Older signal generators, using LC oscillators, have a crystal oscillator that is rich in harmonics. The operator could zero-beat the LC oscillator with a harmonic of the crystal for calibration.
3. A stable oscillator generates sine waves. If you distort the sine wave (clip it) harmonics are generated. You amplify the desired harmonic. Since a square wave has only even harmonics, you would want an asymetric wave for the 3rd harmonic.
4. An LC circuit (parallel resonant) is high impedance, so if it is in parallel with the load, the frequency passes, if it is in series with the load the frequency is attenuated. The series LC circuit is low impedance, so if it is in parallel with the load the frequency bypasses the load and if it is in series with the load the frequency passes.
Narrow band filters are made with crystals, the crystals are used like tuned circuits.