This unit is labelled charge controller. What do I do with it?
What you have looks to be home built and with no documentation - very much in the Maker tradition, but essentially impossible to reuse. Charge control is also dependent on the battery type. Maybe it's a piece of solar history though.
I have a Trace 799 series inverter. I know it is overkiller but should I bring it or buy a smaller unit
I believe that was made by Xantrex, still very active, and owned now by a much larger company. You can contact them for the original documentation. If it is grid-tie, you need to understand how to use it grid independent. A pure sine wave inverter is not necessary for most playa power uses.
What gauge wire runs between the panels and from the panels to the controller
Wire size is determined by Amps and the allowable Voltage drop. On the playa, the distance between the panels and the charge controller, batteries and inverter should be short, so standard size wires determined by NEC tables should be fine. But keep in mind that direct sun heats wires with dark insulation, de-rating the insulation. With just a little reading, anyone can become expert enough on Volts, Amps, Watts and Ohms.
(if you post, your email address appears in the clear, so use a disposable one) Also suggest meeting up with a solar group in your area or with the AEZ group on playa before spending too much money.
The Maker community around Make Magazine is also a great DIY resource for copy-exact projects.