Was going to come back to GW2, but then...
Hard to say what the problem is but the game is working fine for most folks so it is probably your PC or your ISP. Try some of the command-line arguments listed here:
https://wiki.guildwars2.com/wiki/Command_line_arguments
In particular, try these (one at a time):
-repair
-image
-diag
-clientport 80
And post a few screenies of the errors you get.
The ‘firewall’ message is generic — it pops up for all sorts of reasons whenever the game can’t connect, because it’s pretty hard for a program to diagnose connection issues.
Instead, contact customer service by creating a support ticket:
https://help.guildwars2.com/anonymous_requests/new
John Smith: “you should kill monsters, because killing monsters is awesome.”