If you choose Uber/Lyft you will still need to go from MK to a resort or, the TTC. Uber/Lyft cannot pick up at the MK.
Also, guessing you are aware but confirming, if you have entered the MK, you are not eligible to enter another park until 2PM thus why the lack of buses between parks until 1:30.
You can also take the Saratoga Springs bus from MK and get off the Congress Park stop and follow the signs for DS. You enter over by the Christmas store.
Yes there is, by making a connection at any resort. To/from Magic Kingdom, the advantage of a private automobile is diminished by the need to connect also e.g. at the Transportation and Ticket Center.
As mentioned just above, sometimes one of the legs can be done on foot (no pun intended).