Unmanaged packages are typically used to distribute open-source projects or application templates to provide developers with the basic building blocks for an application. Once the components are installed from an unmanaged package, the components can be edited in the organization they are installed in. The developer who created and uploaded the unmanaged package has no control over the installed components, and can't change or upgrade them. Unmanaged packages should not be used to migrate components from a sandbox to production organization. Instead, use Change Sets.
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.
Best Answer
Yes. You are able to edit all of the components that are part of an unmanaged package.
From the Overview of Packages page.