If I've read correctly, you uninstalled Norton from your computer. If that's the case, did you use Norton's own uninstaller program (Norton is so ingrained into your computer that they have to make a tool just to uninstall it properly)? If you didn't download and run the tool, you'll want to do so, as simply uninstalling Norton doesn't revert any changes or permissions it's made. You may also need to dig around and ensure that your network settings (in the computer) are reverted to their default setting. Norton is a jerk.
Regarding the Battleye troubles, my understanding is that it's not even used in Arma 3, and seems to be some lingering code. A quick Google search came up with these same solutions to Battleye issues with Arma 3.
http://steamcommunity.com/app/107410/discussions/1/864973577772340252/http://www.reddit.com/r/arma/comments/2f1y5o/anyone_else_getting_crashes_when_playing/If these potential solutions don't resolve your issue, then we'll need to know a couple things to try and help you further. What OS are you running? Knowing your OS can help us fine tune troubleshooting. It was brought up before, but some of your troubles may be router related. Are you connected to the router with an ethernet cable? Are you connected wireless? What brand of router (or if it's a direct connection to the modem, what brand of modem) is being used? As with the OS, some models of router/modem have well known issues or workarounds, and can help fine tune troubleshooting.
