How do I go about putting standalone PHP files into Drupal 6? I want to add an extra cool PHP Class, where would I put it in Drupal, how do I make absolutely sure the code doesn't present a security risk, should I just make it into a module?
Your best bet would be to include it via a custom module, and to include it would be as easy as using module_load_include().

As for security, I would advise taking a look at the Drupal coding standards documentation, which is unfortunately tagged as D7 and I wasn't able to find the old D6 version, but generally following standards will help prevent some security issues.

Code not written exclusively for Drupal may not specifically work out of the box either, you may have to write a bridge for the code to integrate it better, otherwise you may need to re-write it completely.