"Since then, the "Select Invoices to Print" Task Box has all our invoices form 2004 to 5/31/17 but this last weekend, the latest ones disappeared even though the invoices are in the system."
That means eventually, when someone Prints invoices, the entire queue will attempt to print. Also, having 3000 or more often Crashes a QB program. You need to look at this as Housekeeping and not simply ignore it. You can Print to your printer, which is Turned Off. Let Windows complain about the printer, but QB will simply ask you if everything Printed Fine? Then, you Clear your Printer's queue, which has no relationship to QB.
You are in charge of your own Housekeeping.
Then go back to those Recent invoices and see if they have the ribbon marked to Print Later, or not.