Honestly, Robotnik doesn't seem like he's in a position of authority in this world. He's just some evil guy who lives on top of a spiraling mountain. Given that, it's weird nothing's ever been done about him for his various terrorist acts. Sonic's always there fighting him but the people of Mobius shouldn't be tolerating it either.
Isn't there an episode where somebody doesn't recognize Robotnik and he says something along the lines of "Well, I've conquered most of the known world," or am I just confused? If I'm right, it's probably just an inconsistency (because I know he does talk about Sonic preventing him from conquering Mobius in some episodes), but I like to look at the Robotnik in this series as a deluded supervillain who thinks he's conquered the world, but in actuality all his "subjects" are just too loony and carefree to do anything about him. Read between the lines in some episodes and it really does seem like Robotnik might hold all the strings in Mobian government: in the episode "The Last Resort," it turns out the mayor of the city Sonic lives in is a robot secret agent working for Robotnik. Robotnik's just a crazy guy whose influence stretches farther than you'd think, and the Apathetic Citizens of Mobius just plain don't care enough to prosecute him for his terrorism. That's what makes all of Sonic's heroic acts so important.
I get the distinct feeling that Robotnik and his mooks aren't so inept against people less superpowered than Sonic. Any mortal fool enough to stand up to him would likely be captured and imprisoned in short order.
I wouldn't call the Mobians apathetic, exactly; they all seem to know Robotnik is a "bad guy" to be hated and feared, although they can be persuaded otherwise. You'll even hear the occasional comment about Robotnik's "reign of terror". It all comes down to a combination of general stupidity and naivete, very poor government (the "four leaders" seen in "Super Robotnik" are so incompetent it's pathetic), and that Robotnik and his minions really are capable of harming "ordinary" citizens.
I know Status Quo Is God here, but there should be a real reason why Robotnik hasn't just scrapped Scratch and Grounder already, especially considering how often he threatens to and that he definitely seems to be aware they're only slowing him down. What's worse, Coconuts, their predecessor, is consistently shown as more competent than them, yet Robotnik always gives him the shaft in favor of two bozos he has to know will only fail him again. And is Robotnik really so lazy he can't take care of Sonic himself? He may not be the brightest bulb on the Christmas tree, but he's a lot more effective than Scratch and Grounder, and he has the technical know-how to make some legitimately dangerous stuff.
Scratch and Grounder are probably as good as it gets for Robotnik. Most of the other robots he sends against Sonic are either equally/even more incompetent and much more fragile (like the bounty hunters in the first episode), tend to undergo a Heel–Face Turn (Breezie and Robotnik Jr), or are nothing more than mindless automations. And his organic minions generally leave him.
It's also likely because since he designed them to capture and destroy Sonic, he can't really be rid of them until they complete that task. Robotnik says as much in one episode where he angrily expresses his hope that Scratch and Grounder will succeed in catching and eliminating Sonic as soon as possible, so he can finally dismantle and be done with them.
In the episode "Sonic's Song" the Music Destroyer Robot (appropriately called M.D.) uses his super hearing abilities to find music, but is also super vulnerable to music at the same time.
Well, he was given super hearing so that he could find music to destroy. It's also possible that Robotnik intentionally built him to feel pained by music as an incentive to destroy it.
Why didn't the Music Destroyer from "Sonic's Song" go after Robotnik to stop him from making music?
Because Robotnik's singing was so bad that it wasn't registered as music at all! That's the joke :)
How is it that Robotnik didn't get a full blown Villain Song that didn't have Stylistic Suck? Seeing as how Long John Baldry was a professional singer.
In "Tails' New Home", why are the robots sent to pose as Tails' parents designed to look like insect things? Why not just make them look like foxes so they don't have to put on costumes? Also, what was the point of having the robots dramatically unmask themselves? Why not have them keep manipulating Tails until he's in Robotnik's grasp?
Why is Mobius so bizarre in this cartoon? It's pretty much Wonderland with Sonic and Tails in it. A talking, chilli dog obsessed hedgehog with super speed doesn't seem weird compare to the other things we see on the planet.
Probably the same reason everything is so bizarre in this cartoon. The producers didn't seem to care about being faithful to the games' tone, characters or universe, so they likely just handed some animators and writers who never played Sonic games some concept art and a series bible and said "Have fun!"