|Yes your pc is still connected when you close your browser, eg: Windows or a AV programs are able to update etc. Or you have installed a program that needs to connect in the back ground.|
Only use one firewall on your pc and AV program as more than one will cause conflicts which can leave your pc wide open to attacks.
The only way to stop it all together is to shut down, or disconnect your connection manually.(Note: Logging off is still connected on my router.)
Use a good proven AV program and keep things updated like java, adobe most important is your Windows and AV updates. And do regular scans using AV and malware scanners.
Use a more effective firewall than the Windows built in one. There is also a built in firewall in your router to help.
You can use the following links to test your firewall for holes: