With a clean install of Windows Vista RTM Ultimate (and updates) and no other software installed:
After I install Firefox, both Firefox and Internet Explorer think they are the default browser. Neither one warns on startup about not being the default browser. At that point, I am unable to change the default browser at all.
If I then install mIRC, mIRC will only use Firefox. If I uninstall Firefox, mIRC then uses Internet Explorer.
In answer to your question about your other applications opening the correct browser: most applications simply ask Windows to open a web page, and then Windows itself determines the default browser.
mIRC however needs to determine the browser executable and the DDE topic so that it can control how the browser opens, via DDE if necessary, per the settings in the mIRC Options/Catcher dialog. It does this by looking at the HKCR "http" section of the registry to determine the default browser and DDE topic. If mIRC does not find a valid http entry, it asks Windows to open the web page.