How do I use Organization and Memberships in my site/my profile without setting it all up in AD? Is there a way to have SharePoint store the info?
Memberships aren't administred via AD: site memberships appear in a persons profile whenever that person is added as a site member (so no site owner, administrator, just owners.). AD-groups don't count, the profile has to be added, otherwise it won't work.
about the organization: that is created based on what manager you do have and do have in common with other colleagues. This manager field is filled from user profile import, it does not have to come from an AD import. You could do this manually (blegh!), but you could create some SQL database which contains the organization structure. Connect the SQL database with a BCS connection, and define an import in the user profile synchronization service. Configure that the manager field will get its data from the BCS data source, and your organization will be filled. Note that this doesn't have to be a SQL data source, you can configure other systems, such a HR-system that contains the hierarchy, too.
After the import, you can use the Add-SPProfileLeader to define the CEO's (the person's who don't have a manager).
Have a look at Spence's blog post on Add-SPProfileLeader: It’s all about relevance. The *-SPProfileLeader Windows PowerShell cmdlets in SharePoint Server 2010 SP1 You can create your organizational structure with this power shell command. I don't know what you mean by "Membership" though. Your "memberships" usually get populated with the SharePoint sites where you are in the "Member" group - this is not related to the active directory.