In other words, what goes on behind the scenes when a breakout character becomes apparent? Are the writers handed down an edict from the studio that says "Give us more Sheldon/Stewie/Urkel/etc.-heavy episodes, or you're all fired"?
Well, not "you're all fired," but something like that did happen in the case of the guy on the left in your avatar. Spock was the breakout star of TOS, receiving vastly more fan mail than any other actor (maybe even all the rest put together, but I don't know), and so the network wanted the producers to play up Spock as much as possible. (This is why both seasons 2 and 3 opened with Spock-centric episodes.) But Roddenberry resisted, because he saw Kirk as the star of the show, and Shatner certainly did too. It led to ongoing tension and competition between the two leads, with Roddenberry caught in the middle. Reportedly it was Isaac Asimov who suggested to GR that the key was to play up the Kirk-Spock friendship, to make them inseparable, so that Kirk wouldn't be left behind in the push to keep Spock front and center.
Do the writers suddenly discover that their breakout characters are easier to write for than the other characters?
I doubt that, but it might somewhat be the other way around -- they may discover that a character is easier to tell interesting and engaging stories about than they expected, or that the character takes on a life of his/her own and begins to steal the show just because there's so much potential there. Garak on DS9 is probably a case in point.
Some years back, I had a couple of opportunities to pitch episode ideas to
Star Trek: Voyager, including a pitch for the fifth season. When I was trying to come up with proposals for episodes, I started out with the goal of avoiding Seven stories and Doctor stories, because I figured everyone would be pitching stories about them already so maybe my chances would be better if I offered an alternative by focusing on the rest of the cast. But I discovered that was easier said than done, because there were just so very, very many story possibilities just spilling out of Seven and the Doctor, whereas it was harder to find interesting possibilities for the rest of the cast. After all, most of the crew had either been well-adjusted to start with or had resolved their major issues and conflicts before then; but Seven and the Doctor were both relatively new individuals who still had a lot of learning and growing to do, and both outsiders and misfits who could be sources of conflict more easily than most of the others. So I would say they were easier to write for, in the sense that it was easier to find intriguing and dramatic stories to tell about them, to find new layers of them that could be peeled back to reveal story possibilities. (Although actually writing those rich and interesting stories might have been more challenging and emotionally draining than writing some dull "Harry gets an alien virus" story or something. Good writing is never easy.)
Do episodes that heavily feature breakout characters really generate better reviews and ratings than episodes where they're relegated to the background, or is that just a myth?
I don't know about reviews, but breakout characters are by definition highly popular with the viewers, so it stands to reason that increasing the focus on them would increase ratings (as long as the character is handled in a way that the audience still responds to, I suppose, although actor charisma has a lot to do with it).