After his escape from the hospital he doesn’t even appear disturbed by it.
I think he was too busy being mentally disturbed to worry about his complexion.
Why does Two Face want to kill Gordon's son? How is Dent's ultimate and tragic predicament any fault of Gordon's? The Joker, Batman and the mobsters had more to do with Dent losing Rachel and everything else that he held dear...not Gordon.
He blames Gordon (first a prominent figure within the force, then in charge of it outright) for trusting cops who turned out to be crooked. Wasn't it Ramirez, the female cop seemingly close to Gordon whose mother was ill, who helped the Joker capture Rachel? And he knows that blowing Gordon's brains out won't make him suffer, but making him watch his closest loved one die while lying that everything was going to be okay, mirroring what happened to him and Rachel, will.
I think you have a better case arguing that the Dent-Dawes relationship wasn't believably that strong, but even though I'd say I somewhat agree with you there I still think it was about on par with Bond and Tracy in OHMSS. Speedily established, yet very much plausible and easily accepted once you get past the hasty courtship.
"Please don't kill my boy...Don't...Don't kill my boy...Please". Awsome wrinting, eh? 
At least he didn't say something about what he was going to do with his little finger.
Honestly, it's logical dialogue. Gordon is a regular joe. He's not prone to the impromptu speeches that one could reasonably expect of Joker. I actually think he's the best written (and one of the best acted) characters in the movie.
Dent logically should have killed the Joker in the hospital but that would have not been acceptable by the majority of movie goers.
Yes. And he might have earlier in the film, as we see in the interrogation scene. But by then, he had become warped. Everything was 50-50 to him. Maybe he just didn't have what it takes to become a total vigilante and take the law into his own hands, but he had also given up on his previous "law & order" approach. Let fate decide and his hands would be clean. Justice would at least be served more often than it had been under the Joker's chaotic reign.
Even if I'm speculating out of my

, though, the point is Dent wasn't logical by that point so expecting him to make logical choices is just plain... illogical.
Anyway, some of the issues you're taking with this new "realistic" Batman are like someone taking issue with this new "realistic" Bond never getting killed by the bad guys when they have the opportunity. There's still a healthy mix of fantasy in all this.