I did some testing and here's what I found.
Woocommerce 1.6.6 to 2.0.3 allows the transaction to be completed 100% when customer purchases. Both emails are sent automatically on the purchase. (Customers email for the downloadable product, and your email invoice of the transaction).
They just updated to 2.0.5 which still does not "auto complete" the transaction even when the product is both "Virtual" and "Downloadable".
Something they have changed is effecting our "completed" orders!
But good news is 2.0.3 seems to be doing great.
One thing I found is that if I de-activiated the plugin first, then overwrite the files, it kept my stuff as it was, just upgraded/downgraded to the version I was overwriting. But when I DELETED the plugin, then stuff got very wonky. So deactivate, and overwrite with Filezilla if you go that way. (Then simply re-activate in the wordpress plugins).
Again, do your backups before proceeding with it ;). (Full backup, + the CPanel partial backups to be safe)