Seeing as no-one else has had a go at this, allow me:
1)Keep private certain sections of the website from competetive teams
Any and all sections of the site can be kept private. When users Register they join a group called registered. You can then create a new group of users called "Private", and add the users you know are on the team to this group. You can then assign "module access rights" to the relevant modules to only this group of people.
2)Be able to assign updatability to certain pages to different people in the organization. (sub-admins)
Same as above, only create a list called "SubAdmins". You then need to assign "module admin rights" to those modules to this group.
3)They want to create a mailing list.
You can send mail to any group from the backend. You need to be an admin to do this. This is strictly one-many. If you want an e-mail discussion list, you may be better with other software. There may be mailing lsit modules for XOOPS - I've not used them.
4)They want to be able to moderate membership, to keep competators from accessing the private pages.
You can require admin validation of an account before it becomes active. This would allow you (as admin) to control access. In this case, if only your team are to be registered, you can omit creating the "Private" group in step 1 and simply use "Registered" instead.
5)they would like to have a photo gallery (private).
See answers to 1,2, & 4
6)Blog and forums capability.
Not sure about "Blog", though there are modules to do this - check the modules repository. There are several good forums for XOOPS, CBB & NewBB are worth looking at. You can again lock down the permissions on these so only registered people can use them.
7)They want to tie into cafepress.
Can't answer - sorry.
8) Web Calendar.
Again, there are several avaialble, and I can only advise to play around and see which one best suits your needs.
One problem here is that I believe the Documentation has been taken down at the moment as it was hacked (could be wrong), but this system is pretty simple to get to grips with. If you do a google search for "XOOPS tutorial" you'll find some useful hits to get you up and running.
You made the right choice having XOOPS installed, it's not perfect, but it's pretty darned good for most things and should easily be able to do what you want.
Hope this helps,
Gareth.