There was only one Buck Rogers serial, but Flash Gordon flew into space three times to battle the Satanically majestic Emperor Ming.

Question Number One has to be:
Why did Emperor Ming keep coming back from the dead?
Ming was the prime villlain of Flash Gordon's newspaper strip. Creators Alex Raymond, and Don Moore used him as a common foe to rally support for Flash, Dale, and Zarkov when they traveled to all the different nations of Mongo. Although there were many minor nuisances, and a few worthy adversaries like Witch Queen Azura, the Sax Roemer-like Emperor Ming was tailor-made from pulp novels and newspapers of the 30's -- the age of Mussolini, Stalin, Hitler, and Franco. If repeated defeats weren't fatal for Ming in William Randoph Hearst's Journal-American, he wouldn't disappear from the silver screen either!
Reason One: Ming was nearly as important as Flash Gordon.
Charles Middleton brought such prescence and dynamism to his portrayal of Ming that he dominated the entire trilogy eventually. It might have been true that he was tentative at first, but he was confidently wicked and resourceful by the second adventure, and swaggeringly domineering all through the third. In retrospect it is obvious that Middleton was the best actor in the entire series, and splendidly focused the attention of fellow cast members, listless writers, and excited audiences -- all of whom thrived on seeing their heroes prevail against such a charismatic antagonist.
Reason Two: Ming was somehow impervious to fire.
Flash Gordon's Trip to Mars repeatedly demonstrates this special power of Ming, as if answering Flash's remark: "We saw him perish in those flames!" -- Referring to Ming's apparent immolation in the Temple of the Great God Tayo, near the end of the first adventure.
Reason Three: Ming faked dying, with the help of his henchmen.
Inexplicably, Ming had allies who shouldered some of his dirty work. It made some kind of convoluted sense that if Flash could escape from the disintegration ray on Mars, then so could the cunning Emperor.
Zarkov immediately suspected Ming for the Purple Death dust, after spotting a Rocket Ship above the Earth in the third adventure -- even though Flash protested about seeing him dead. After they flew back to Mongo, Dale, Flash, and Zarkov learned their old nemesis was ruling the planet again, with only a rueful nodding of Prince Barin's head, instead of an explanation.

Question Number Two can be anything, but I choose:
Why did Flash's second serial take place on Mars anyway?
The Science Fiction genre had a long fascination with our next-door neighbor since the days of Percival Lowell's canals, and H.G. Wells' first invasion-from-space novels. Interplanetary Space was like Rafael Sabatini's Spanish Main, or Zane Gray's Wild West in the so-called Space Operas. Planet Mars was an important location in the mythos -- much like Port Royal or Dodge City in other genres.
Answer One: Flash Gordon competed with other Space Operas.
If Flash stuck around in one place, he might have lost some fans. For instance, rival Buck Rogers rode around the Solar System like Fran Striker's Lone Ranger in the newspapers. Flash Gordon never left Mongo in the newsprint adventures until after WWII began. Azura was Witch Queen there before Beatrice Roberts portrayed her as a magical Martian tyrant. Some Universal episodes didn't mention Mars at all, and there are a number of bloopers in the dialogue which may indicate an executive decision to send Flash to the Red Planet might have been made after production started.
Rumor: Universal tried to cash in on Orson Welles' War of the Worlds.
Mercury Theater's Halloween broadcast, and the hysteria caused by some fake news flashes during the radio drama, caused a major media frenzy in it's wake. Baby Boomer fans long assumed this notorious event was the cause of Flash temporarily relocating to yet another planet, but there is decisive evidence to the contrary.