circle navigation
This is possible on a flat earth. This is a map used for demonstration purposes (it is not accurate):

Now, imagine a plane travelling along one of the inner circles from Europe to America, the long way round. And by the way, I think the word you are looking for is 'circumnavigation'.
Deep tunnels, such as mine shafts, we would loose our minors out the bottom which im pretty sure people would know about.
No mine shaft on earth is that deep. Indeed, the deepest mine shafts ever built were not even 4km deep, and at its deepest points the ocean floor is about 10km. In fact, no-one has ever drilled deeper than about 12km. In other words, this would never, ever, cause a problem.
That's two simple errors. I now request that you aplogise, as you have made obvious mistakes, and clearly are getting this information second hand or from thin air.