Patrick is correct on how these traits and mechanics interact with one another.
Unfortunately, there was a misprint on Tanglevine - it is supposed to give the targeted creature the Unmovable trait as well. This text was present on all previous print runs and throughout playtesting, but somehow missed the final prints. Therefore, the creature cannot be Pushed out of Tanglevine's grasp. Sorry for the confusion, I am going to post an erratum for this.
Unlike a Push, Teleporting can affect Unmovable creatures.