If I fork someone else's private Github repo into my account, is it going to appear in my account as a public repo?

No. You can fork it and it still remains private.

Private collaborators may fork any private repository you’ve added them to without their own paid plan. Their forks do not count against your private repository quota.

https://github.com/plans

When you revoke access for a member on the main branch (master repo), it will yank the forked repo as well. So, if you added members to a team, and they forked from the master repo, always ensure that you either have merged their changes or you have a copy of their changes before you remove them from the team, as the members forked repo gets deleted when he is removed from the team and you wont have a way to fetch his changes (if only you care about those).

Here's GitHub's answer:

https://help.github.com/articles/what-happens-to-forks-when-a-repository-is-deleted-or-changes-visibility/#deleting-a-private-repository

Deleting a private repository

When you delete a private repository, all of its private forks are also deleted.