"Why did Batman become old and withered when he stopped fighting crime 8 years back?"
"How did he suddenly become fit again to play around when he was barely being able to walk?"
"How did Bruce Wayne's broken back get repaired by suspending him from a rope? "
The purpose of the limp shows the distance Bruce has fallen from the empowered physical prowess of being Batman. And as it is never focused on as a major plot or narrative point, a quick resolution with one of Bruce’s many expensive gadgets is a minor point in the more significant focus of the first act of the film: Bruce’s mental and emotional state — something that is manifested physically in his limp. If anything, it shows Bruce’s initial reluctance to heal, mentally or physically.
Additionally, the /Film writers criticize Bruce starting out low, then rising, and repeat, but this analysis ignores a major focal point of the film. Alfred worriedly remarks to Bruce that putting on the cape and cowl doesn’t automatically enable him to be Batman again. He’s afraid Bruce lacks the belief (in contrast to Bane’s abundance of belief — something else pointed out by Alfred) and conviction to become the Batman we know, love, and aspire to be. And he didn’t. So as he is thrown into the pit defeated and broken mentally, physically, and emotionally, the successful climb out of the pit is the first time Batman truly emerges in the film.
It’s a major convention in film for characters to heal significantly faster than they might otherwise in reality. It seems silly to have this suddenly be a big criticism after a century of film.
"Why did the entire police force go underground?"
"How did 3 almost 5 months pass by and no one did anything?"
GCPD has been established as consistently lackluster in the first two films, as well as specifically under the leadership of Foley. Mindful of the eight year gap, the police force is unprepared and unaccustomed to handling real threats. Foley consistently underestimates the significance of Bane’s attacks (note the dismissal of the Stock Exchange assault and the discussion to retract his men from chasing Bane to instead follow Batman). Confronted for the first time with by a notoriously menacing and dangerous man, Foley reacts impulsively. To corner Bane within the sprawling network of sewer lines, a large number of officers must approach from every possible entrance. Bane’s intelligent, and more than this, he can predict different people’s psychological reactions, and he used this to his benefit. Under Foley, a leader for peacetime, the police really were that inept. As evinced in The Dark Knight, Gordon would never sacrifice his entire force with such haste.
As for the CIA, the plane they used was neither large nor particularly sophisticated or militarized. It’s not entirely off base that they didn’t notice the second plane until it came closer in its approach (note the turbulence and the pilots’ rattled expressions). Yes, the agent broke protocol by ushering the hooded figures onto the plane without identification, but he wasn’t one to play by the rules anyway, i.e., his quickness to torture and threaten death to get information. The logic here isn’t perfect though, I admit.
"How did Batman reach back to Gotham after being imprisoned in some god forsaken hellhole?"
It’s important to note that the /Film time-frame criticisms seem inconsistent. Sometimes Bruce is gone for weeks, sometimes it’s months, but now it’s only days from escaping the pit to returning to Gotham? Bruce had around three weeks to return after leaving, and because Wayne Manor isn’t in the actual city of Gotham, he probably returned there, aka to the batcave. Thus, he’d have had access to the Batsuit, and seemingly endless gadgets to relatively easily sneak into the city. Given the resourcefulness he’s proven to have had over three films, finding a way back into the States is probably relatively straightforward. Hell, he could’ve called Alfred to hook him up.
Considering he has an intimate knowledge of the bomb and it’s workings and he followed what happened to Gotham on the television Bane setup, if Bruce has any ability to read a calendar he’d have a pretty good idea how much time is left on the bomb. He’s Batman and a master detective. Why is using that skill unnatural for the character?