I'll try not to be an arse. It's hard.
I am running 10.9.3 on an older xServe. My first problem was getting the bluetooth to speak to the petrel or predator (I have both). Turns out, the dongle driver was the problem. The IR dongle for my Uwatecs is just as bad, and you can't tell which part isn't working. I run firefox instead of safari. Can't help with a Safari issue. The shearwater software installed perfectly, but the computer firmware must match the firmware in the software. I had a friend borrow my predator, and he had it about 6 months, doing the firmware updates every time they had one. I got it back (I never do firmware updates) and my predator wouldn't speak tot he Mac. So, you have a lot of places to go look, but I'd make sure my bluetooth was working first (have your computer speak to your phone), then make sure you have the latest release from Shearwater Desktop, then make sure the latest firmware is in your petrel. If it all crashes down, give Lynn a shout either here or by e-mail. She answers these questions all day long.