in answer to your question about some sort of 'shared communal calculus database' across institutions globally...(Am assuming your are not around the corner from me here :0)
I really am not sure about that....great Idea! But I have not been involved in anything of that nature....perhaps others might know....I suppose if you consider something like GIT, and think of a database in those terms...then I guess it ought to be possible, for example, for me to share a database of questions with you-even if you were in outer Mongolia! (As an aside-I recommend Tim's advice about STACK, in the very least, to enable you to share questions across your institution).
So, sorry not able to give a more accurate answer on that front. That said, to understand the database module, I think-tis probably built akin to that of microsoft access/filemaker databases...
a couple of links for you then...with the aim of being helpful-if nowt else!