Sharing a database (Family) with another user.

It depends on if you need to be have other devices quickly updated or not. (And I emphasize need, as it points to one option over another.)

It doesn’t have to be instant. A few minutes delay, even update once every hour will be OK.

But does it require updating while someone is off the local network?

That will be ideal. Now, using Evernote, I can share a notebook with her, she can open Evernote on her iPhone, scan (Take a photo of a document) and save it to the same shared notebook. Is that possible? Are there any disadvantages?

Only the disadvantages inherent in any remote connection: poor network quality, slow / unresponsive servers, bandwidth throttling / limiting maximum concurrent connections.

If you want a remote sync solution, you should start a common account on Dropbox for you both to sync to. (This cannot be done on iCloud, as it’s only for personal syncing.) You essentially have one account and you will both use the same username / password and store name. This way you will be able to sync to the same remote location and update each others’ machines. Obviously, any one else you want to involve in the syncing would use the same sync info as well.

Is it possible to keep one (ie.: Family) database on Dropbox and use the method you described and have others in iCloud?

Just to be clear: You can’t have a database on Dropbox and you should never put a DEVONthink database in any cloud-synced folder.

But… yes, you could use a common account to sync a family database with Dropbox and also sync your own databases to your iCloud account.