I think I've figured this out. I believe I need to add a flag for each data set (so Admissions_Count would be 1 in all admissions records, 0 otherwise) then merge all the datasets on all of their common criteria to get a flat file that can be a single datasource for tableau. I'll work on this...
The millions of rows are probably coming from improper joins. You should be able to join all of the tables together using ID. Just make sure that you use the appropriate join (left or center).
If you can link your data sources before bringing into Tableau, of course that would be great. As said above, that can be more of a data issue of how your links work.
However, I definitely understand if all of your data needs to remain separate and won't join as you need. I've created a lot of dashboards with multiple separate data connections and filters that work on all of them. Here are some of the things I've learned in setting these up:
If you want to join data sources on a single field, best advice is to make sure they all have exactly the same (linking) Dimension field name, and the values in those fields are exactly the same in each. It will make everything much easier.
For filtering, actions work well (and it's good to have a few actions set up to highlight the same department in each view, for example). But I did a lot of speed tests, and parameters are by far the best way to apply filters to multiple sheets in an entire dashboard.
The ideal scenario for parameters is if you have a relatively fixed list of values in the filter - for example, if you had a total of 20 Majors to list that wasn't likely to change much. If your field has hundreds of values, and is rapidly changing, you likely have to consider other options (probably Actions, or different filters, in your situation).
The other current drawback with parameters is they don't allow for multiple selections without predefining all of these combinations. I say current, because it is an idea you can vote on http://community.tableau.com/ideas/1313
But, if you want to select the same College and/or Major in all of your views no matter the data source, I put up an example, with a sample of how to do it here:
For every field that you want to filter on, make a parameter - here it seems like College, Major, Academic Period (maybe ID too, depending on whether you want to filter). Show the single parameter control for each on the page. As long as it's set up right, this is a great way to control multiple data sources, and apply the "partially-universal filters" that so many have been asking for!
Please let me know if this doesn't work out for you, I'm happy to help more if needed!