Starting from the bottom: To show the Like button in the K2 Content Module, you should simply have to enable the parameter in that module that's something like "Enable Content Plugins". That will cause the JFBCContent plugin to fire off and do it's work.
For your other issue, can you post (or private message) the URL of the page your seeing this on? It sounds like the Like button is for the 'current' page, not the 'destination' page, like it should be.. therefore, the Like button is shared across all the different categories on that page. When you click into a category (the destination), that's a different URL, so the Like button isn't showing the same count as on the previous page. We should be able to help you fix this so each Like is for the destination (individual category) page, instead of the current, but it's best to see what's happening first.
As for why the Like count is different when you change the menu item, that's correct behavior. When you change the Menu item, the URL for the page will have a different URL (usually the Itemid=xx portion will be different). This is a different page, and Facebook will assign Likes to it separately than other pages, even if they're similar.
Hope that explains things,
Alex