Not a lawyer, just a frequent international traveler.
No one can answer your question about cost without knowing your nationality because jurisdiction matters in the fee schedule for passports.
You should replace a damaged passport: Any damage other than very light and normal wear on the edges. You do not want your passport scrutinized for fraud, alteration, etc when traveling.
Usually after transitioning from minor to adult a person must apply for an adult passport as if they never had one. It's usually just the case that they want to verify your identity in person. source but this source is American and judging from your post, America does not have a 'HM Office" and you are probably not American.