Is my "two room, two switch, four lights" diagram correct?
Yes, your diagram is correct. It will achieve what you described.

You should always connect the ground, it is a safety measure. While it may not appear to do anything during normal operation, it's presence covers many failure cases that have been discovered the hard way - death by electrocution, house burning down, etc.

You should also absolutely check your local electrical codes and regulations for how to correctly do an installation, there are many things that are not obvious or intuitive. In the USA, "NFPA 70" is the name of the code book. There are also many laws that cover who is allowed to do this kind of work, inspections, and etc.