What sites are accessed on the internet during installation?

None with the installer.

Since I already have the latest Java installed, which site connection attempts can I block and still have installation success?

Once you downloaded the installer, you can block anything you want on the Internet, and don't need to install Java on your PC. Sweet Home 3D installer provides its own version of Java (that's why the size of the installer is much larger that the Java Web Start version).

If so, does Java need internet access after Sweet Home 3D installation?

Still no access to Internet is required, except for links that point to Sweet Home 3D and other web sites in the help and about dialog. You can also download furniture libraries with your browser and install them without any access to Internet.