I have seen, in numerous screenshots of Dogecoin-QT, a "dig" pane (equivalent to "mine" in some others): However, my client has no such pane: Does anyone know where that option has gone? Was it removed in an upgrade?
Yes, it was removed in an update. The developers have decided to hide it as a way to discourage users from wasting CPU time. CPU mining this big of a coin will get you no where.
It's suggested to mine using an external mining program like CGMiner or practically any miner recommended for Scrypt. Check out Litecoin wiki's Mining Software list and look into Scrypt mining guides such as the Dogecoin wiki's 'Beginners guide'.