The damage to the fruit looks to be from codling moth. I believe the black mound you see in the core is fass (i.e. insect poop). The larva of codling moth bore into fruit and start eating their way into the core, leaving behind a trail of frass. Later on the caterpillars exit the fruit, pupate, then emerge as adults to complete a second life cycle. For more on codling moth, see: https://pnwhandbooks.org/insect/tree-fruit/apple/apple-codling-moth 

The skeletonized damage that I see on the leaves is not rust, but may be caused by the apple and thorn skeletonizer or possibly the pear sawfly. It is hard to say without seeing the pest. The apple and thorn skeletonizer will typically cause the leaves to roll up. Monitor the leaves of the tree to look for either yellowish-green caterpillars with black spots (the apple and thorn skeletonizer), or small slug-like larvae (the pear sawfly). For more on both possible pests, see https://pnwhandbooks.org/insect/tree-fruit/apple/apple-apple-thorn-skeletonizer and https://pnwhandbooks.org/insect/tree-fruit/cherry/cherry-cherry-slug-pear-slug-cherry-sawfly-pear-sa...

Finally, to answer your question about drinking cider from the fruit. I myself have consumed lots of cider from fruit with codling moth damage, and I lived to tell about it. However, the food safety aspect of this question would be best answered by our OSU Master Food Preservers. The toll-free hotline at<personal data hidden> is open from 9 a.m. to 4 p.m., Monday - Friday through October 9, 2020.