We use Deploy on Demand to move files from TeamSite onto the webservers, but how do we get files into TeamSite that are not dcr's?

If the files are not in TeamSite
Navigate to the location you which to put the files in TeamSite. In the FILE dropdown menu, there is an option called "Import files..." Selecting this the first time will cause a small program to be installed on your computer, say yes when your computer asks you if you wish to install it.

The top half of this screen is your local computer, find the file[s] or directory that you want to import and select it. Click [Add] or [Add All] then Import and the files will be sent to TeamSite.

Once files are in TeamSite
Once files are in TeamSite and you would like to move them to your local computer to work with them, simply select the file or directory and choose EDIT > Download. To move the changed file back into TeamSite, use EDIT > Upload.

Note: This will not lock the files, it is highly recommended that you use EDIT > Lock and Get Latest before downloading. Also note that if you do not change and submit the file, you must unlock them before anyone else can use them.