Any node can “rehost” the content by simply requesting the content. It would be permanently “rehosted” if that content was also pinned.
That’s correct but it’s really isn’t specific to IPFS. If you host some content on a web site and I download it there is nothing you can do about preventing me, short of legal action, to prevent me from making that available on my own web server. IPFS just makes the process automatic.
That probably depends on what you mean by “uploads a file onto the private network” and “removes it locally”. If you add the content to node A, no other node requests the content, and you remove the content then it’s gone. If any other node requested the content before removing it from node A then it will still be at that node and accessible to any other node on the network.
If your private network were a cluster the cluster manager would coordinate unpinning across the cluster.
I’m not quite sure, so someone would have to confirm, if unpinning it and running gc would guarantee that the content was removed from the local node.
Hopefully I got that right and it helps.
You might want to look at this thread How can I be certain that my IPFS node is not hosting content X short of `rm -rf ~/.ipfs`?
In addition to the gc you’d also have to make sure it’s not on the MFS