I almost feel like there's a missing serial between Trip to Mars and Conquers the Universe. How did Ming get back to Mongo? For that matter, how did Barin get back to Mongo when Flash took his rocketship? Who the hell is Ronal and where did he meet Flash? What's this business about "returning his sword?"
As noted, Trip to Mars fits seamlessly onto the end of the first serial, but in this case they took no care for continuity. Maybe with all the cast changes, they figured hell with it.
Anyway, we're left to fill in the gaps ourselves. I assume Flash must have met Ronal during the first serial while we weren't looking, maybe after the Tournament of Death and before returning to Ming's palace for Feast Day. In the comic strip, that's about where Barin was granted rule over Arboria--though in the comic strip we don't meet Ronal for several more years; he's Barin's cousin, and incidentally, is in love with Dale.
Ming's escape from the disintegrating room isn't all that complicated--I figure he left his robe to be disintegrated, then escaped through the same hole Flash blew in the wall a few episodes ago! He returned to Mongo in the same rocketship in which he came--and then, after returning to Mongo and seizing back his throne, launched a devastating attack on Mars using the Z-O ship we'll see later--blasting away most of the Martian atmosphere...nitrifying the soil...boiling away all the oceans...leaving the planet a dead, oxidized, barren desert...
As for how Barin got off Mars, that has me stumped. Maybe when Ming returns to his throne, Aura sends a rocketship to Mars to find and retrieve Barin. So Barin comes to Mars, bringing along Mighty Toran as his aide (am I the only one who noticed that Barin's aide in the third serial is named Toran?).
As for Flash, Dale, and Zarkov not being surprised that Ming is alive, maybe they had their doubts all along--after all, he tricked them once before. The guy's like a cockroach, you just can't kill him. He's worse than Scorpius on Farscape. Yes, when Barin confirms that Ming is alive, Flash is mildly surprised, but remember they assumed Ming was the culprit from the moment they first spotted Torch's rocketship dropping the death dust.
So there's my theory.
Am I way too big a Flash Gordon fan?
Actually, the continuity from serial 1 to serial 2 was not quite seamless. Flash, Dale, and Zarkov departed Mongo in the one they had used to get there from Earth--Zarkov's invention. The first series ended with them over Manhattan about to land.
The second serial also shows them about to land--but they are in a different rocket ship! The one that they are now in looks like one from Ming's fleet. Also, they apparently overshot NYC to land in a farmer's melon patch.
When they fly to Mars Ming refers to the rocket ship as one that they stole from him on Mongo. Of course Ming cannot be trusted to always (or ever) tell the truth but he seems to be right. I can only think that the writers forgot the details of the final chapter of the first serial.
One other minor discrepancy is that the opening segment has the radio operator on Earth communicating with the rocket ship using CW. In previous communications (e.g., when asked to shut down all electrical generators(!)), voice was used.
Now that I think about it, though, (and I think that I will then stop thinking about it), the radio discrepancy can be resolved by saying that Zarkov's ship had a voice transmitter and Ming's had a telegraph-type key.