How to make your profile private?

You can't, everything on the fediverse is very public. Even if such checkbox existed, all admins still have a copy of everything in their database.

I don't see this functionality in Lemmy. Mastodon has settings where you can limit your posting visibility to only those who follow you. But anybody that you don't expressly block will still be able to see your profile. More privacy features mighr be in development.

Probably your best bet is to keep your fediverse handle as anonymous as is possible. That would be the best advice I have to give. Also, don't give clues as to your whereabouts in your posts. Nothing on the Internet is trult private and infosec is hard. Always assume that a fediverse instance could be compromised at any time. Use some common sense and basic operational security and you should be okay.

You can't. Just have separate accounts.

If you don't want other people to see your posts and comments you shouldn't post and comment on a public website. Duh.

Make a new account for every post.