How about the issue of the number of turns in each coil?

Is it better to use wire that's small with many turns, or wire that's big
with fewer turns?

Number of Turns

The number of turns will have a direct effect on the coil's d.c. resistance. A large resistance
will decrease the current if the power supply is not changed. But if you build
the coil first, and choose the voltage of the power supply second, then you can achieve any current
you want. So I don't see a problem with resistance; just concentrate on the coil and projectile
first, and then choose an electrical source that can provide both the voltage and current you
need. And select power output transistors that can handle your voltage and current and power.

Just keep in mind a very basic, very simple rule: The coil's magnetic field is directly proportional
to the number of turns (actually turns/inch), and to the coil current.
Really! It easy to maximize performance: just keep piling on turns, and increasing the current.

That rule is why I ended up with fat coils on my coilgun. I could add turns more easily than
I could make a bigger power supply.

This whole exercise of designing a coilgun is

Figure out how to store as much energy as possible in the magnetic field around a coil.

Then figure out a projectile that can couple with the field as much as possible, and has
the minimum mass possible.

When you put the projectile near the coil, the system seeks the minimum energy state. That occurs
when the projectile is in the center of the coil. So the system dumps a bunch of mechanical energy
into the projectile so the whole thing can finally reach that minimum energy state.

Anyway, back to the number of turns question. Which is also related to the wire size. I haven't
been able to identify all the variables and equations that need to be optimized. This is all
such an iterative process. But I think it's really a matter of maximizing turns and maximizing
coil current, until you run into some practical limitation. Here's a few limitations that occurred
to me:

1) Current density inside the wire. Once this gets too high, the coil can't
dissipate the heat fast enough. You can get around it a little bit by reducing the duty cycle,
but eventually the wire melts during a single shot. For that reason, thicker wire is better.
And that's why big motors have thick wires. This should lead to a discussion of the theory and
design of thermal dissipation, but I won't go into it.

2) Power supply voltage. There can be problems switching high voltages. I used
2N2955 for switching transistors, and they're only rated up to 60 vdc. So that puts an upper
limit on my power supplies. You could use other switching devices with much higher ratings. For
example, an IGBT (insulated gate transistor) has a maximum voltage of 400 or 600 or even 1200
vdc.

3) Power supply instantaneous current. The only economical way to supply huge
current is from big capacitors. (Or perhaps a car battery?) There's a limit to the price of the
capacitors you can afford. And if you want portability, there's a limit to their physical dimensions.
And the physical dimensions are a trade-off between the capacitance and the wvdc (working voltage
dc) rating. By the way, one researcher has an article "optimizing a capacitor-driven coilgun" in
IEEE Transactions on Magnetics. I need to find a copy of that someday.

4) Output current. There are limits to managing high currents. The 2N2955 are
rated for 15A continuous current, or 150W total power dissipation. You could use another device
with higher ratings. For example, the IGBT is intended for electric motor control and can handle
a lot more current.