1) IMO, an iron core would be damn heavy and inefficient.
2) That sounds right. Also need a spark gap
3) Probably. You need to be able to determine the frequency of the primary by measurement. There will always be some parasitic component you won't include in your calculation.
4 ) Yeah, that's the whole point. The main reason a telsa coil works is because it operated an resonance, which builds up the voltage much higher than it would otherwise be. The primary need to be tuned to the resonance of the secondary, after that has been measured. There is a good you-tube video on how to measure the resonate frquency and tune the primary. Sorry, I don't have the link; you'll have to search for it. Also, it's my understanding that there is significant inter-coil capacitance, so your calculation using a lumped cap. as the top cap will by off by a little. It's very important that you measure the resonant frequency.