While it's Played for Laughs that Paddington mistook Mr. Brown's age to be "80", it's not entirely because he's being Innocently Insensitive. Given bear life spans are different from humans, he may have been saying Mr. Brown looked good for someone who was 80 bear years. Especially as the opening titles of the film mentions them.
Paddington is still a cub. Why was he tried as an adult and sent to a prison with adult men?
Because they thought he was an adult in human years?
Actually, the film seems to go out of its way to tell the audience that Paddington is not a cub. He tells people that "Aunt Lucy did so much for [him] when [he] was a cub" (the implication being that he's no longer one) at least three times. If Paddington were human, he'd probably be 18 at the very least and I'd guess more early-twenties.
In the climax, why would the carnival organ make a "wrong" buzz when Phoenix enters the passtune incorrectly? It's supposed to just appear as an ordinary organ, only revealing its nature as a safe if the passtune is played. Additionally, it buzzes as soon as a wrong note is played, which would make it vulnerable to Password Slot Machine attacks. I guess it's just Rule of Funny.
Given that Paddington only wants to buy a pop-up book for Aunt Lucy to show her the sights of London, why does nobody suggest to him that he buy a cheap new kids book from a high-street store rather than a one-of-a-kind antique for £1000?
Partly because the gift is for a beloved family member, and recommending something cheap would probably not go over well. Partly because they don't want to break his heart. Mostly because it's Paddington Bear, and amazing things tend to happen where he's concerned.