it trims ... to relieve back pressure on the elevator
Yes.
If I'm correct in my assumption as stated above? If so what is the big deal behind this feature on the Airbus?
There is no deal at all.
Assuming the THS is trimming a nose up attitude because you are pulling on the sidestick, wouldn't the reverse also be true
that with a nose down input the THS would trim nose down?
Yes.
Back to AF447 and the THS being full nose up. If the pilot would have just inputed a full nose down on the sidestick, wouldn’t the THS trimmed nose down too?
Yes.