Ive always had a similar problem on I think all my Ibanez's. Always just my G string though. Bend it, it would go flat, whammy it, it would go sharp. I stripped out lock nut screws, used graphite on the saddles, you name it. Its always boiled down to finding that sweet spot before I locked it down. The fine tuner had to be just about exactly in the middle, in that, same amount of turns up or down. The G string itself, I always had to tune it way flat before locking it down, which......after locking it, would bring it just about to 440 with minimal adjustment on the fine tuner.
This seems to always do the trick. It happened on all my 6 string Ibanez's in the past, and then on all my 7's from past to present. Ive always just written it off as one of those weird tweek things.
As far as the trem goes, I only have that problem when I first pick up the guitar. Its either flat or sharp. I whammy it a few times, and run a few scales, bending notes & stuff before I start playing. This seems to settle it back in. On a few rare occasions, I have to give a slight turn to the tension screws one way or the other if its been sitting awhile.
I would agree with the springs not being properly stretched.........or even perhaps, add another one.
But thats just me