Concerning the Zygons plan to invade Earth by hiding in the paintings. If the Queen and the Doctor(s) knew what they were doing in 1562 why didn't they just lockup/remove/destroy the paintings so they could not get to Earths future?? The Doctors even took the cube device with them but left the painting with the queen??
The only reason the Doctor(s) found out what was happening back in 1562 was because of the 11th Doctor being summoned to investigate what had broken out of said paintings in the future/present. Altering that would've prevented UNIT from getting the 11th Doctor involved in the first place, and worse, it would actually have altered his own personal time line.
Plus, the Moment would have selected that particular future to show the War Doctor, because it would lead to an event where both the Zygons and humans come off better due to the his future selves involvement, and most importantly his own contributions to said outcome.