Most tuning issues (particularly on a new guitar) can be traced back to the nut or the tuners. The issue I see happen most commonly is some of the strings being tuned to pitch downward (meaning the note is sharp and you let out some slack to get it in tune), which leaves a little bit of slack at the tuner and can also cause windings to catch on the nut. The result is that when you whack the trem with the nut locked (or even just playing in general), some of the slack gets pulled out and then the tension on the string between the nut and the bridge is uneven, and causes a lot of weird tuning gremlins.
Another thing to check are your springs. I've had springs very slightly sliding in and out of the block when you're going at it, and then when you return to zero, the springs end up at different depths; all of this happening on a very small scale but enough to cause anomalies. In that case, consider replacing them.
One last thing to mention... I've had new springs that aren't completely broken in and as such, contract unevenly when you use them. In that case, they need to be broken in. One way I've done this is to play with just two springs in and whammy the fuck out of everything, then swap in another one or two after a few weeks.
There's a slim chance the issue is the 800lbs. gorilla (inferior knife edges, etc.) but hopefully you can debug it outside of that.