Welcome to ETO, daveclifford95!
As to your questions:
1. Yes, it will. And it's a so-called "tap" on the coil (or a signal "pick-up") for propagation. Also its length will be a factor in a frequency shift. But the trimmer's range (2-22uF cap) should allow for adjusting the oscillator frequency into the FM band. What will have the most significant effect on oscillator frequency will be the coil's (inductor's) actual, physical construction (number of turns, diameter and length), as noted in the circuit description.
2. The pot is an audio input volume control.
3. I'm sure it could but for your first effort, I'd avoid "improvements" until you get this one working correctly.
4. Without knowing, for sure, what the inductance of the coil actually is (or the trimmer's value at the preferred resonance, for that matter), you'll be hard pressed to determine the tank's resonant frequency mathematically. If it were me, I wouldn't bother at this point. Once you've successfully created a working circuit, with a resonant frequency within the 88-108MHz range, use that frequency (however you determine it (I'd guess from an FM radio dial/readout) in the formula for resonance to determine the inductance (while guessing at the value of the trimmer) for the inductance of the coil. It'll be off a bit due to the effect of the antenna. And the 3pF cap will influence the resonant frequency but, again, it should not exceed the influence of the trimmer.
Now, with all that said, note carefully the parts layout for the circuit displayed at the cryptomuseum page. Within the oscillator portion of the circuit, relative component position will have an enormous effect on its operation, especially concerning the arrangement of the trimmer, coil (plus tap) and cap. It'll take a lot of "tweaking" of those positions to get to all work together properly. And make sure your fingers are nowhere near the circuit when you test for the resonant frequency (the noted "hand effect" in the circuit description).
Good luck and have fun.