It seems like you're deliberately misrepresenting, but I'll explain anyway because I know that some people might be confused.

The Mastodon app is a client on your phone which accesses servers in the network.

The network consists of multiple servers that are interconnected to each other. Content from one server is automatically cross-hosted to other servers when it is discovered on those other servers. That's how Federation works. I know it's probably an oversimplification of how activitypub works, but it's generally good enough for most people, and the important part is really that content is present and visible on other servers.

When you sign up to a server your account is stored on that server, the posts that you make are stored on that server, as well as automatically cross hosted to other servers which have people following you.

If the owner of a server pulls the plug for whatever reason the content on that server will no longer be directly accessible, if your account is there you will lose your account. The copies on other servers will remain as they have been copied. The rest of the network will continue operating without that server and the accounts that were hosted on it.

About asking whether or not an instance owner can sue instances they don't like, that sounds like absolute nonsense and I'm not even going to bother trying to understand whatever point you're trying to make with this.