Assuming it's Windows 2k or XP (which it sounds like)...
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1. Whenever I connect to the internet, I get a yellow notification box that appears from the taskbar telling me that my internet connection is now active. How can I turn this notification off? I've done it before, but I can't remember how.
Go into the properties of your connection - There's a tickbox called 'Show icon in notification area when connected - this will get rid of your icon for the connection itself, but should stop the box appearing. Other than that, you may wish to play around with the general Notification settings.
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2. When I want to disconnect from the internet, I close all internet explorer windows, and an auto-disconnect box used to appear, where I'd have the option to either stay connected or to disconnect. That box no longer appears for some reason, so I have to disconnect another way. How can I get the auto-disconnect box to appear again?
This one will be in Internet Options - go to the Connections tab, select Settings for your connection and then Advanced - i do believe you need to tick 'Disconnect when connection may no longer be needed'.
HTH