You're getting 2 issues confused. Firstly, you're running a firewall on your host PC. Secondly, you don't connect to a firewall, you connect to a proxy. Those settings in your IRC client for specifying your firewall type are irrelevant for most home users. They are there to enable people to connect if they can only connect through a proxy. The description 'connect through firewall' is misleading as well. Really it means 'Connect through proxy'.

To change your hostname on IRC, you either need to be an IRC operator (different from a channel operator), know a friendly IRC operator, or have a bouncer. There is no way to do it without one of thase 3 things. The easiest way is to get a bouncer. They are relatively cheap, and pretty straightforward to set up and use. Mine only costs me £4 a year. However, the first 2 options are free