Problem 2 - If you use Set Focus just before moving the image to where you want to go it will try to cut where you are asking it to.
If it still is not accurate, you need to run the Calibration routine.
Problem 1 - All wood is a bit different from piece to piece and even within the same piece. Proofgrade has less of this but no matter what, the differences in weather while the tree is growing will make differences in the wood. If the wood is pinned in place and you have not moved it and you have changed nothing to move the image, you can re-run the cut and it will repeat the previous path.
When a piece is highlighted a widget will appear in the lower right corner, which if you pick it will show both the exact location and size of your piece. It is these numbers that decide the exact place on the tray that will be cut, even when the image and other things say differently.