I'll take a shot:
1. Are you asking whether your group should think about joining the national PTA organization as opposed to just starting up your own local independent group? If so, my suggestion would be that you carefully consider the pros and cons of joining national - you can get access to a lot of resources, including help setting up your local organization, but it may also tie your hands in some ways that you may not want, including having to pay dues. Check out their web site:
National PTA Ithink you can go slow on this - I suggest that your group talk with other PTA and PTO groups to see what their experiences have been.
2. PTO = Parent Teacher Organization. It's a generic term for local orgs that are not affiliated with the national PTA. See:
PTO vs. PTA: What's the Difference? - PTOToday.com
3. At my kids' schools they called this type of organization "The Home and School Association." Be aware that your group will have to have a unique name, especially if you are thinking about estabishing it as an official not-for-profit separet from the PTA. The easy way to do that is to work the name of the school, or school disrict, into the name, such as: "The Anytown Middle School Parent and Teacher Organization."