My 5.5 month old male cat is persistently asking to go outside...?

He just has to get used to not going out anymore. It usually takes about 2 weeks maybe a month at the most. But one thing if your gonna keep him indoors only then do not let him at all. Cats are very regimented in their ways whatever there used to thats what they want to do. If he's used to going out he'll want to go out. But you can break this you just have to stick to not letting him out at all. My cat was an outdoor cat for a year till I took him and made him indoor only. It took a little while but he got used to it eventually. Your vet is absolutely right about waiting to neuter him. They can do it younger but there is definitely a higher chance of something going wrong. My brother's cat was neutered at about 6 weeks old and died a week later from it so I wouldn't do it till at least 6 months to be safe.