Time dilation

Special relativity, worked out for a constant speed, also shows time dilation in accelerating frames (eg. Twin's Paradox) and Gravity is an acceleration. In general relativity, it is considered to be a difference in the passage of proper time at different positions as described by a metric tensor of spacetime.

 

I don't think so - but I may be mistaken, as I only know GR in approximation. According to the equivalence principle, locally the observed effects from acceleration are the same as the effects from gravitation.

For example, the observations inside a rocket in deep space that is accelerating at a proper acceleration equal to g and with two clocks at different "heights", should be the same as in rest on the surface of the earth (assuming uniform g).
The usual example calculations do not include time dilation and length contraction effects due to velocity because they are done for a small change of velocity relative to rest.