2
MYSQL DB - Check, XOOPS currently only supports MySQL
Python not required - Check, XOOPS is written in PHP and a little JavaScript
Caching - Check, XOOPS uses SMARTY
Complete French localization - Conditional Check, I *think* French is at least translated for the core
Various permissions - Check, XOOPS has usergroups with many settings to access etc.
Powerful templating - Check, XOOPS uses SMARTY
Thought to work with thousands of members - Check, xoops.org has currently 16 thousand registered members
Short URLs - Conditional Check, there is a way to do this. Check the ~200 post thread regarding shortURL's
Powerful forums - Check, currently being expanded
Links directory - Check
Downloads directory - Check, being expanded atm
Both with unlimited subcategories - Check
News with categories, images, comments, RSS feed,
and trackback ! - Conditional Check, what's trackback?
Ability to add/edit/remove fields in the members profile.
Ability to choose a registration form depending on the membership type.
This is a little tricky to do with XOOPS, but I'm told that there are people working on making this a viable option
Pending registration (submit membership) - Registrants can be set to require administrator activation
Hope XOOPS is what you are looking for