Can you get directly from one park to another by bus? Or do you have to go back to your resort and catch a bus there?

You can go from park-to-park, but you cannot go from DTD to a park.

in addition, all park-to-park buses involving MK is at the TTC (and there's no Epcot <-> MK bus since the monorail is there). i've learned to bus to the Contemporary and walk over to MK when i have to hop to/from MK

There are no buses from Typhoon Lagoon directly to any of the major parks.