Great questions...here's some feedback.
1. Maybe not. If your school and your members view the PTC as a committee of the school, then your group would not have its own EIN. Your group would need an EIN only if it is operating as a separate organization. However, no harm done. If there's no $ associated with the EIN, there isn't anything to report to the IRS.
2. No. It seems you could open a separate checking account using the school's EIN (with the school's blessing), and manage it yourselves, even though organizationally, you're a committee of the school. As long as your money is in account(s) opened with the school's EIN, regardless of who has signature authority to the various accounts, your money belongs to the school.
3. If you mean, can you take advantage of tax exemption for purchases using the school's id, but setting up your group with its own EIN, no. Can't have it both ways. Your PTC and school really have to decide whether your group is an extension of the school, or an indenpendent organization.
4. Assuming you act as an independent group, with your own EIN, then the IRS expects you to apply for 501c3 if your annual gross receipts are normally greater than $5,000. If you have $10,000 in gross receipts for a few years ("normally"), then you should apply for 501c3. Once you are a 501c3, you must file the annual form 990EZ.
As your group matures, you'll want to have that important discussion about independence both with your school administrators and your members. While many, many parent groups are independent of their schools, it's not the only successful model. In your community, at this point in your group's evolution, maybe it's best to stay under your school's wing. That's for you all to decide.