Uhm... I haven't played with this, so this may be a dumb question -- but is the height of the coil directly related to anything else? Or is this a case of "Sure, put the antenna on top of a 2' tall stick and you're done, but of course that has no effect on how well it works."?
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I think the height of the secondary is a mechanical function of the number of turns, the wire diameter and any voltage insulation requirements necessary to avoid arcing between turns.
Like yokozuna said if you can scavenge the transformer you can easily keep it under 100, you can purchase magnet wire on Ebay for under 30 and cab radially make hv caps on the cheap https://www.youtube.com/watch?v=khduXAjvktU&feature=youtube_gdata_player
That depends on the quality.
You can buy one here. A
A lot of it depends on your specifications and how much you salvage. If you get the transformers from something like an old tv, that leaves the biggest item to collect as the magnet wire. If you are good at scavenging, you should be able to meet your $100 budget pretty easy. If you're buying all new parts, you probably won't.