No, there's probably no reason why you can't go to the Principal and offer to help with any of those types of projects. In a couple of schools in my area, those specific projects have been done by the student government and/or groups of teachers, without any help from the PTA. If money needs to be raised to complete any of them, the Principal might send you to the PTA--or the Principal might have some funding available--or they might ask you to help write a grant proposal to get money for the school.
A school could have a PTO and a PTA, but it would be highly unusual and probably too confusing. The PTO would have to have a very separate and distinct purpose from the PTA. I am aware of two schools in my area that have started nonprofits separate from the PTA, because they needed to raise money for a very specific and ongoing science and technology related project--it's run by parents and teachers, but doesn't have PTO in its name and doesn't do anything else other than raise money from corporations/foundations for this particular project. In that type of a case, it does make sense to have the two organizations. There's absolutely no overlap between these nonprofits and the PTAs at the schools.
Hello,
I want to become involved in my kids school. For example: start a recycling program, create an outdoor classroom, start a garden, etc. Do I have to create a PTO to do this? Is there a reason why I can't go directly to the Principal to complete these projects?
Going thru current PTA is not an option. If I do need a PTO, can a school have both a PTA (national org) and PTO?