Hello Hannah, and welcome to AO.

The first thing to remember is that even if you can trace a computer and its connectivity, there is no way of determining who is actually using it or its connection. A good example of that would be if your computer was hijacked or "owned" by a third party. They could use it remotely to indukge in all sorts of criminal activity and it would look as if it were you.

To trace to your home address and name would require the assistance of your internet service provider (ISP). In most countries this would require a court order or warrant, so only law enforcement and national security agencies could do it.

As for putting websites on a host or signing petitions; the host site will see the IP address that you are coming from. In my case that might get you to within 70 miles of where I live.

If you use a proxy, then all the site will see is the proxy server's address. If you use a public computer like in a library or internet cafe, then all they will see is that address.

I hope that answers your questions