Is there a way to search for "completed" tasks for all projects on a particular date?
With Todoist Premium, you can view all completed tasks (incl. all completion events of recurring tasks) in the "# Completed" history in the top-right corner. You can choose "Any time" from the time filter drop-down menu and will be able to see all completion events across all projects sorted by day.
Feel free to take a look at this short video: http://www.youtube.com/watch?v=IWCBVE3IhJ8 to see how it works. This feature is shown at the 0:45 mark.