Your problem sounds quite annoying.
Here are the solutions I suggest for this:
Delete this WiFi configuration from your phone.(You don't need#0011##. Just tap on that WiFi connection in Settings.) Reboot the router. Double check the router setup to see what type of security is enabled. For the initial setup, at least, make sure the SSID is visible. Recreate the connection on your phone.
Press these menu options on your phone: Settings > Wi-fi> (Choose your network by tapping on it for a few seconds) > Modify network config
Check "Show advanced options"
Choose IP Settings: "Static"
Enter these details
IP - 192.168.1.253 or 126.96.36.199
Gateway - 188.8.131.52 (I chose 192.168.1.1 since that's my router address)
NPL - 24
Netmask - 255.255.255.0 (I could not find this option)
DNS 1 - 184.108.40.206 or 220.127.116.11 (I had the latter by default and let it stay)
DNS 2 - 18.104.22.168
And save. It should connect.
A third solution that might help in addition to doing the above includes turning off a couple of Wi-Fi settings. Go into the Advanced menu from the Wi-Fi settings on the S III and uncheck Network notification and Auto connect.
Open the Phone app and dial *#0011#. This opens a special “ServiceMode”. From here, tap the menu button and go into WiFi and then hit the ON button to switch it to OFF. Hit the back button twice to exit the screen.
I hope one of these solutions works! If not, let me know and I'll keep on looking for a plausible solution