That is correct.
The light leaving the gravity well of a star is red shifted. If the star is moving towards us then that red shifted light will be blue shifted.
No, it is not false.
Not sure what you are trying to say here. According to the bible light was not omnipresent, but since we are in the physics section I have no idea why this would come up in the discussion.
This is incorrect. If photons were pure energy (whatever that could be) then they would have no other properties, such as spin, frequency and momentum.