How awesome!! Thanks for sharing. I can't believe how tight those courses were. Me and Koji would've failed. Honestly, if my dog was slipping that badly in the weaves, I wouldn't have continued that many times. Just not worth their injuries. I hope none of the dogs were injured on that flooring.
As for shoes. I run on grass or dirt (and used to run on rubber mats in training). I wear trail running shoes for dry days. For wet/dewy days, I have a pair of waterproof hiking (semi-running) shoes that do the trick (they have gore-tex). I also don't really run in the rain, so it rarely gets "that" went.
I have seen people wearing golf shoes (soft spikes) and baseball shoes (rubber cleats). For me personally, I'm afraid of the golf shoes, I golf and know what kind of traction I get. I also know I'd be the idiot who'd twist her ankle because I didn't pick up my foot properly. I'd wear cleats (grew up in them), but I know my feet would hurt after awhile, so I go with trail running shoes.