If I use Admin Tools to create a .htaccess file the site downloads the index.php file instead of serving it. What setting causes this, why does it occur, can it be fixed, and what are the implications of fixing it?
An .htaccess file should be compatible with the server in order to work and to not create issues.
Admin Tools .htaccess maker mentions that there is no guarantee that the produced .htaccess will work at your server. There are so many possible server configurations and there is not necessarily a particular setting that may cause specific side-effects and certainly with the information given in your question it's hard to tell - as we don't know both what your .htaccess looks like and also how your server is configured.
Either build step by step your .htaccess to see what's causing your issues - or collaborate with your hosting provider on the issue.