Method to deal with conflicted copies among group in Dropbox

I haven't used their service, but CloudHQ seems to offer what you need.

From their website about the Basecamp / Dropbox integration:

CloudHQ simplifies management of Google Docs, Basecamp, and Dropbox files Manage all your Dropbox and Basecamp files directly from a Google Docs interface CloudHQ will continuously synchronize Dropbox, Basecamp, and Google Docs CloudHQ can also backup Basecamp projects and Google Docs to Dropbox

You can't set permissions for shared folders in Dropbox.

This feature has been requested for three years, as has received almost 150k votes in Votebox, but Dropbox hasn't implemented it yet.