Both of those are big ones for me, too. Especially since they tend to come and go, depending on the needs of the story. Manufactured weaknesses in general tend to get old quickly... like EVERYTHING being yellow in old Green Lantern comics.

It always bothered me when people talk about Batman being a "normal guy without powers", and thus is more appealing or requires less suspension of disbelief than superpowered heroes. As if unparalleled fighting skills, PhD-level intellect and knowledge in multiple disciplines, enormous wealth, advanced preparation for every possible scenario that would be classified as delusional or paranoid in our world, and enough will power to fuel a dozen power rings is any less fantastical.

I've always thought the same thing. Batman is my favorite DC character, but he is in many ways less realistic and harder to believe than Superman. Superman is plausible since he's superhuman in a reality where superhuman powers exist. His speed alone would allow him to be far more productive than a mere mortal. 

Batman, on the other hand, couldn't be who he is in reality simply because of time restraints. Even if someone had the capacity to be Batman, the training needed to master each of the fields of knowledge that he's mastered would each take a lifetime. It makes much more sense that a character like Ra's al Ghul, who's lived for centuries, would have all those skills. 

Rick Jones! What the hell is so special about Rick Jones that apparently a superhero cannot lay eyes on him without wanting to make him his sidekick?

It's one of those things that doesn't really make sense but I also think is kind of cool. The lantern doesn't function in the third dimension (matter) but in the fourth (time).

I think at the end of the day where a story carries the most power is in the immediate, uncritical reaction. Before you have a chance to think about how goofy it is the emotional impact of the story claws at you. Suspension of disbelief. If you lay it all out like that then it comes off as ridiculous, but the immediate impact I get when I see Batman do something is the impact of seeing a mortal man elevating himself to the level of a God, while no matter how much they try to emphasize Superman's humanity I always see him first as an alien, and thus distinct from me and much less impressive.

Theres also the matter of the character profile you've outlined not being impossible. Superman defying gravity is impossible. Completely incompatible with physics, the core language of the universe. But a guy who is a martial arts master, a genius, a polymath, a billionaire, paranoid and exceptionally strong-willed is possible. Incredibly unlikely, but possible. Most of the things you've listed are all linked to each other. If someone is a genius and exceptionally strong-willed then odds are they'll be able to learn martial arts and multiple sciences and be able to acquire wealth. To say that Batman, as a person, is impossible is to say that genius, star athlete and business magnate are all mutually exclusive, which is of course untrue.

I think it helps that in most media, such as the animated series and the Nolan movies, Batman's intelligence is reigned in. He's a brilliant detective and he's smart but he's not a multi-disciplinary genius. To most people Batman really is just a guy with some genetic gifts and expansive resources. And in the end, as exceptional as he is, he is mortal. He can be shot and killed, just like us, which makes everything spectacular he does much more impressive. And the fact that he doesn't have powers, that he doesn't (as Uncle Ben would say) have great power and thus great responsibility, yet he does it anyway because it's the right thing to do, that's just extraordinary. When a character is a man who is forcing himself to be a God it's very hard not to empathize with him.