If I delete a file from my storage at dropbox.com, but it still exists locally on my computer, will syncing cause the local file to delete?
Yes. It's easily tested: after a few seconds any file deleted through the website will also be deleted from your computer's Dropbox folder.
On a Mac, it will also not be moved to your Trash, but really be gone. Of course, it can still be recovered using the Dropbox website though.