Specifically, will I be able to search across *all* task notes, including task notes for completed tasks?
unfortunately, Todoist doesn't have an option to search through task notes, but we will consider adding such feature in the future. You're welcome to vote for this request in our Votebox: http://todoist.com/Vote/showProposal/355/
With Todoist Premium, you'd be able to search through task names. That would include both active and completed tasks.