Without knowing the network and what exactly the channel services are on that network, I can't be too specific. But, in general, you have 2 different services. The first is handles nicks. You register your nick and log into it when you connect. This lets you keep others from using the nick. That is what you are doing with Q. The other service is the channel service. This service controls who gets automatic op status as well as other things like automatically kicking people or voicing them. To change who gets the op status, the channel service needs to be updated. This would be done using a /msg command or a specific command followed by what you want done. For example, you showed how to register and authorize yourself using /msg and also using /auth as a separate command. A similar thing would be done to set up who gets automatically opped.
Now, if you are getting automatically opped while not logged in, but losing it when logged in, then that means that logging in changes your host info. In some networks, being logged in will change something like mynick!ident@myhost.com to mynick!ident@users.undernet.org (for Undernet). Most likely, your real host information is entered for automatic ops, but not your logged in "host".
In general, any kind of questions regarding the network's services can be answered more quickly and accurately by joining #help or whatever channel the network uses for help. Often, a question can be answered in a few minutes there rather than potentially hours here. And they already know what services they use, while we probably will not.
