This is a self-test, not a certification. The goal is not to defeat the verification goal, but to learn something. So yeah, it's perfectly acceptable that the tests are not bullet-proof.

Depends on how the broken program writes to the log.

If it does

    while true; do echo hello >> bad.log; done

Then renaming bad.log will not solve the challenge.

There are ways to cheat but not so simple; there's a script that checks for the solution and a hash of the script is checked for modifications.