I'm setting up a new computer, and need my DB files. Can I just directly transfer my DB folder between computers or will I have to re-download several GBs from DB on the new computer?
Copy your Dropbox folder to the new computer and rename it to to Dropbox_OLD or similar. Install Dropbox on the new computer like you normally would. When the installation is finished Dropbox will immediately start to sync your files from your account online. Pause syncing or exit Dropbox. Move the content of Dropbox_OLD into the newly created Dropbox folder. When the move is complete, and not before, resume syncing or re-launch Dropbox.
At this point Dropbox will begin indexing all of your files. This process will take a while, especially if you have a lot of data to go through. During this time it may say that files are uploading or downloading, but it's only transferring comparison data and any changes that it find. Be patient and LET IT WORK.