By the sounds of it, your sub subdomain maps to the same directory as the main domain and all other subdomains (www and other, etc.)
However, if all requests to the sub subdomain are internally rewritten to the /sub subdirectory then you can presumably just create another .htaccess at /sub/.htaccess in which you set DirectorySlash Off - this then applies to all requests to the sub subdomain. (Assuming you don't also access the same subdirectory via a different hostname. You can prevent this if you wish.)
The /sub/.htaccess file is also where you would be implementing #3 in your requirements.
The root .htaccess file simply rewrites all requests to the sub subdomain to the /sub subdirectory.
Aside: Since you are asking this question on ServerFault it is generally assumed that you have full control of the server. In which case it would be preferable to configure this sub subdomain in its own vHost container that points directly to the /sub subdirectory (or somewhere outside of the main domains directory tree ideally). In this case, you would not need to implement #2 of your requirements and you just set DirectorySlash Off for the entire subdomain.