Problems moving to production site
Exclude /sites/*/files/css and /sites/*/files/js from repository. It's just a cache so it should never be committed or migrated.
Before each migration purge /sites/*/files/css and /sites/*/files/js. That way you will get rid of months old cache. After each migration, make sure these directories are writeable. mmtwln seems to be a string added by Drupal or some module to prevent proxy caching, or it may be a string added by a proxy itself. it shouldn't affect static files at all (except cache prevention, but even that isn't sure)
Your site fails at saving cookies. Check your cookie domain settings in sites/mysite/settings.php and make sure you have right data in example.org/admin/config/system/site-information


If above is insufficient:


manually (via ssh/sftp/ftp) compare offending CSS files. Does the version downloaded from server look the same as version you was sure you uploaded? If not, consult your sysadmin as your server is malfunctioning somehow.
if the version you download from server via ssh is the same you uploaded, but differs from the one you downloaded via http, check carefully your .htaccess - probably there are some redirection rules that make your server to serve file from different location. Common offender is boost module, if you used to have it installed and now only it's rewrites are left, you can easily get the exact behaviour you described.