No repro given the steps you outlined as demonstrated in this video:
You're right in that if you delete a test that is holding the elements referenced by coded step in another test, it will cause compile failures. My suggestion in such a case is to copy the element definition out of the Pages.g.cs file before
you delete the test and manually add it to coded step of test 2. You won't use Pages.PageName.Element format, but use your own code once you've extracted it out of Pages.g.cs.
For the problem you're having with renaming a test or moving it into another folder, can you send us a project that exhibits this behavior for us to study?