1) I can't find the dielectric breakdown voltage for any of the gases I want to test! 3) Where in the world can I get these gases for a price within my budget?
What gases do you want to test? Maybe we can help with where to find some preliminary information and suggest some possible sources of gases who might help out a science fair project participant.
2) How can I seal the spark gap in a container?
A lot depends on the voltage involved and what the pressure difference is. Have you had a high school chemestry lab where you bend and manipulate glass tubing? If you have, then you can probably adapt some methods and put a small diameter solid bare wire through a glass tube and heat sealing one end and then placing the glass tube through a rubber stopper. Hot glue is something else that can be tried. The goal will be to use insulating materials and spacing.
Also, the article says...
In order to generate sufficient voltage to create a spark, the secondary coil should have orders of magnitude more turns than the primary coil. A ratio of 750–1000 is a good place to start. In order not to exceed the power rating of the current-limiting resistor, the ratio of turns (secondary:primary) should not exceed 1000:1.
Now does that mean for every 75 (or 7.5) turns I have in the primary coil I should have 100 (or 10) in the secondary? Is there a minimum number of turns?
A ratio of between 750:1 and 1000:1 is a good place to start. This means for every single turn on the primary, you need 750 to 1000 on the secondary! Have fun winding! You need enough primary turns to make a good enough electro magnet to pull the normally closed relay contact open. To minimize the number of secondary turns you need, you want to keep the number of primary turns small.
BEWARE: There are still be a few minor issues with the parts list and proceedures that have not been corrected yet.
Using the bare (uninsulated) 12 AWG copper wire in the material list for the primary is wrong. When it is wound on a metal bolt, it WILL short circuit the coil making for a poor electro magnet and a poor electro magnetic field for a transformer. Type TWHN solid 12 AWG copper wire or similar should be called for if this is to be used for the primary winding.
The relay contacts are shown incorrectly. The normally closed contacts should be oriented away from the bolt head so that when current flows, the primary and bolt act like an electro magnet and open the contact. The normally closed contacts should be wired as shown, it is just the mechanical orientation that is wrong.
The description of the building of the apparatus should indicate that the Neef vibrator should be constructed and tested and be in good working order BEFORE winding the secondary! This should be done to allow working with the circuit without the high voltage danger!