He went through this crazy journey of getting inside the criminal mind and training with the League of Shadows to become this ultimate weapon against the filth of Gotham only to try and throw it away at the mere prospect of being with a woman. Twice. Then he gives up being Batman after only being in action for, what, a couple of months?
....No he didn't. He stayed with being batman until he thought he was done. Also, you misunderstand his mission. He only wanted to be batman to pear away the corruption and then kickstart the police into action.
Why blame Dent's death on Batman? Is anyone going to believe Joker didn't do it? It's not like he didn't target Dent once before.
I don't quite remember the rationale for it when I looked up this very question online, but I remember being satisfied with the answer. However, this has little to do with his character arc. This is a plot quibble that is tangential to it. Either way, once the police were able to mobilize after having cleaned up Gotham, his job was done. He just thought he'd be able to walk away from it clean, but the plot decided to make it a harsher break. It might not be the best reasoned plot point, but it's not something that is vastly different from how his arc was going to turn out. It's more like it just cemented that he would really have to give up everything, even his good name, to fix this city once and for all.
So 8 years later he dons the costume once again because...
...someone took his money. Which he never got back and then Bane beat him near to death.
Pretty sure that he smelt trouble brewing with bane. The money being exhausted only tipped him off.
Fell for another woman that got pretty cozy pretty fast (which also almost gets him killed) and once again gives up being Batman the second the opportunity arises AND HAS A NEW WOMAN who robbed him and then led him straight to Bane and also kills "Bruce Wayne" so he doesn't have to deal with that anymore. The whole thing was stupid.
You keep mentioning 'fell for a woman' as if this is a bad thing. Bruce of the TDK saga craves intimacy and love, which he was never able to recouperate until he felt he honored his parents enough to fulfill their dream of a clean Gotham. Once he did that, he was open to persue what he always wanted, a life of love, but Rachel died. That's why he is broken down for years, he gets attached and doesn't let go. Narratively, this is a good thing, because if you are in a situation where you need to be a grizzled stoic badass to survive and you are a grizzled stoic badass, then you've nullified the conflict. Nolan's Bruce is a sap, and he leads with that as his thesis, so which makes how he deals with his situations more interesting because it's an actual transformative experience. He is a badass, but he's a big softie at heart. And I'll gladly take that over BvS's batman whose arc is composed of something a literal 5 year old could figure out.
Which makes his ultimate romance with Selena Kyle decent too. Yeah, she is a criminal, but he sees the good in people to a fault. This usually doesn't work out, but in Selena's case, it did and not only does he live a better life for it, so does she. They improve each other, having her open up to companionship where she is unable, while Bruce has someone to have his back and shoot people if he needs it. But then again, I am a sucker for opposites attract pairings, so feel free to disagree with me on that one.
I think a more apt comparison to Batman is Black Panther. Panther basically goes off a photo that no one bothered to check the validity of as reason enough to try and kill Bucky.
I mostly agree, but I have more sympathy towards momentary anger than a grudge going on for years. Batman had a year and a half to figure out all the shit he needs to about Superman. Black Panther literally just had his father die, a more intimate and personal connection than a boss with his employees and strangers on the ground floor, literally in his arms against an enemy wherein investigation may literally not be a realistic possibility due to the lethality of the Winter Soldier.
Batman had ages to figure out who superman was and is maintaining his anger based on an event that happened so long ago. Black Panther's wound is raw and new, and he could have very well been the only one able to stop him as far as he knew.