Single Blog / Multiple Websites / Category-Theme Website Specific

I have several websites which are divisions off of a single company but have distinct themes for each website. I would like to set wordpress up to serve a single blog to all of these websites with the following criteria:

Main website would show all general and division categories.

Division 1 website would show general and division 1 categories. Categories from other divisions not available.

Division 2 website would show general and division 2 categories. Categories from other divisions not available.

etc…

Also as far as feed syndication, I’d like to be able to prepare specific feeds for the individual sites only containing categories shown at that site.

I would also like to have each site use its own theme so the blog is branded to each site it is on.

I had thought about doing a different blog for each site but then I lose the ability to share the general news items between sites.

Anyone have an idea of how to do this or if there is a plug-in somewhere I might be able to use to accomplish it? I did a search but didn’t find anything to fit a multi-site/single blog scenario. Seemed like everything I found did the opposite.

Thanks in advance for any assistance!