First I would strongly recommend to remove that log from Pastebin and the link from your post! We can't do much with it (we are not Microsoft). But more importantly it contains private data of yours and sometimes of your contacts, which could all be abused by people.
You should never post such logs in public! Only give them to Microsoft when you contact
support.
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Anyways,
Like the troubleshooter suggests, start by making sure MSIE is online!
Using FireFox (or any other browser) as your default browser hasn't got anything todo with all this. You use FF to browse the web, but many programs use MSIE components to connect to the internet. The same with Messenger, it needs MSIE (or rather its components) to function properly.
Think of it as MSIE being part of your OS and thus it's a needed component. That you can also use it to browse the internet is just a 'bonus'.
If you get an error, the first place to look is the official help and support:
http://help.live.com/ > Table of Contents > Error Messages > Windows Live Error Messages >
Error 81000306
You can also look at one of the official blogs:
http://messenger-support.spaces.live.com/ >
Windows Live Messenger sign-in errors >
Error 81000306
...just as said in the top post of this thread.
Note: if they tell you to visit a certain site to check out certain connections, you best always do it with MSIE, not with any other browser, as MSIE uses special components to 'talk' to sites, the same components Messenger needs and uses.
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