1. the outlook connector pack refers to the outlook you have downloaded. So it is for hotmail.com and outlook.com both, they are the same thing.
2. I know about desktop alerts, but not sure about sounds. I turn all windows sounds off, but I would think it were possible. These are all settings that you may have to find. Here is how to turn on desktop pop-ups: Turn on or off new message alert pop-up window - Outlook - Office.com
3. not really. Outlook 2010 should be able to do a whole lot more than mail 2012, though they may be very similar. Never used thunderbird.
You can repair outlook if it is giving you trouble. This won't change settings: Repair or remove Office - PowerPoint - Office.com
^If that doesn't work, try testing your email once more and let me know if you still get those errors. Unfortunately I will be leaving shortly (vacation, yay). I don't have access to my computer with outlook to tell you more exact details on where everything may be found.
PS there is also a way to make emails come as soon as outlook starts up, and you can change how often it checks for emails, and stop it from checking for emails when you close out of it. That is what I have done on mine.