Is it not possible to create a Time saved search and grab the Invoice in which the time entry was billed? Also, it seems that's the other way around is also not possible? A transaction saved search showing the Invoices and the billable time. Also, is there no system notes joins in the Time Entry saved search? Is there a way to determine when a Time Entry is approved in a saved search?

I would think the only join would be from the Invoice back to the time entry. The time record is pretty limited and does not have all the same things a normal transaction would Have you tried Applying transaction, applied to transaction, posted time or billing transaction field joins? Those seem the most likely if at all The best way I've found to track time entry approvals is using a custom workflow and recording the approval date in a custom field on the time entry when it enters the approved stage