There are no buses between DTD and any of the parks or the TTC. Using Disney transportation, you need to get to a resort to catch a bus to DTD. From the MK, that's actually fairly easy as you can walk or take the monorail to the CR and catch a bus there, or get to the Poly or GF by boat or monorail and do the bus from there.
As to the cab question, yes, there will be cabs available at the TTC to take you to DTD if you choose that method.