Does the Terra Mystica app implement the official rules updates?
Yes.

The Terra Mystic App (on Android at least) includes the changes to Cultist Power and the variable starting Victory Points.

It does not block the Spade tiles in the last two rounds.

The above is based on my experience playing the app as of August 2017.
Edited to reflect recieving [5 points for founding town/4 earth cult for a Spade] as my final round token in a game in late August.