How to get Drupal 7 installed in a sub-folder / sub-directory properly?
When I do this, provided I have a clean install, I do not modify anything. Take the domain http://diw.ifi.uio.no/, where Drupal is installed in both /www/, and /www/gruppe01/, etc. (I just create the number of subdomains I need for various experiments). My DocumentRoot is set to be /www/. I just direct my browser to http://diw.ifi.uio.no/ for the main site, and  http://diw.ifi.uio.no/gruppe01/ for the first subfolder site, etc. and it works - just like that.

Have you tried starting with a clean install and just install Drupal in the subfolder using the standard install procedure without changing .htaccess? (While not always necessary, I recommend that you have $base_url set correctly in the site's  settings.php.) 

In my experience, Drupal works fine out of the box when installed in a subdirectory of a vhost. The fact that there is another Drupal in the DocumentRoot does not make this any different, as long as each Drupal site has its own .htaccess, it will not try to read the .htaccess in any superior directory.

Examples:


http://diw.ifi.uio.no/
http://diw.ifi.uio.no/gruppe01/