When the transformers are on, they create RF signals that run along conductive paths and light up glass tubes full of gases. All of my components are hanging upside down underneath the paths/tubes in a 19"x24" area....maybe 5" max gap between components.
OK, so, you are deliberately creating RF fields, and 'broadcasting' them to your gas ...
Have you checked the 'range' of your transmission?
Reducing its power to the minimum is going to be helpful to you, and appreciated by the neighbours!
Improving your 'aerial design' (like putting a ground 'plane' {maybe even non-planar} an appropriate distance behind it) should improve its directionality, and allow you to operate with minimal power. Then you can benefit from shielding your aerial feeds and rf generator circuitry, and also from ensuring that your RF generator isn't putting RF back out onto its electricity supply.
Regarding the Arduino and its sensors and power supply. Yes, anything {ie everything} vaguely conductive and unshielded/ungrounded in the neighbourhood is going to act as a receiving aerial and pick up your RF transmission. So you need to have everything, cables, the lot, equipped with well-grounded shielding. (But naturally, you don't want anything accidentally short-circuiting to ground.)
And because people will also act as receiving aerials (hopefully harmlessly, see below), I suspect that any contact sensors may need to be appropriately filtered - in hardware, software, or both.
Did you check the Wikipedia entry on plasma lamps? Plasma lamp - Wikipedia
The radio frequency field produced by plasma lamps can interfere with the operation of touch pads used on laptop computers, digital audio players, cell phones, and other similar devices. Some types can radiate sufficient RFI to interfere with cordless telephones and Wi-Fi devices several feet away.
And that Wikipedia entry also carries a worthwhile reminder about the potential of a risk to Heart Pacemaker users - if this thing is ever likely to be exposed to the public (and vice versa) there needs to be some sort of warning displayed.