Question on that….I can run any Max algorithm on a core unit, I understand, but the Max has to be present, correct?  Or can I actually save patches to the core units I buy?

No, the Max doesn't have to be present and you can save presets with all the algorithms on your Core.  As long as you have a Max registered to your Eventide account, you'll be able to use all of the algorithms.  If you resell the Max to someone else, then you'll no longer be able to.