1. How do I make a category a "Hidden Category" ?

Put the magic word __HIDDENCAT__ on the category page.

2. How would I implement this "site notice"?

Edit the 'system' page on your wiki called MediaWiki:sitenotice which pop up a short message to user's when they visit the site which they can dismiss once read. Remember to remove when problem fixed. Note: only admins can edit it. (the "MediaWiki:" name space contains all sorts of system message pages and other functions used to customise the site)

3. How do I get the site notice to show back up one it's been dismissed?

You would need to update you're Wiki's mediawiki:Sitenotice id, and it should show again. Or, try clearing cookies/cache, that should make the solution for just yourself.
A. What exactly is the id for?
In theory I would venture a guess as to say the MediaWiki:Sitenotice id is to keep track of how many sitenotices you issued, if your sitenotice changes and the id does not then you will not see the sitenotice so you would bump the id up +1 with each new sitenotice change (thus able to see how many different sitenotice messages you have given out over a span of time)
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