Just navigate to that installation URL and it should install fine. I don't know where you saw that you can only install packages that are listed on the AppExchange. In the original sandbox, when you made the package, it created an installation URL on each version. Just navigate there and sign into the other sandbox.
Oh, substitute "test" for "login" in the installation URL. That way you can put it into a sandbox and not production.
Yes. You are able to edit all of the components that are part of an unmanaged package.
From the Overview of Packages page.