An observer A at rest in a gravitational field has some amount of time dilation relative to an observer B far away from the gravitational field. Would the clock of an observer C in free fall at the same location as A be dilated relative to Bs clock exactly the same as observer As clock (i.e. at a given location in a gravitational field, does the amount of time dilation at that location depend on whether the observer is at rest or in free fall)? Does a free falling observers clock become increasingly dilated relative to a clock at rest at infinity as it falls in a gravitational field (in addition to any time dilation effects due to relative velocity)?
There will be no effect at all at t=0. When t grows, the object will gain spatial speed, and this will contribute to an almost negligible time dilation. The greater effect (at least near a planet or stsr and not a black hole) will be an increasing gravitational time dilation as the object falls deeper into the gravitational potential.
So the answer to your first question is a rotund «No.» and to the second «Yes.»
You can read my article «Sistemas en caída libre y,corrimiento hacia el azul gravitacional» ( Spanish - «Free falling systems and gravitational blueshift ), Lumen, September - October 1993.