You need to pose those questions to the church's Financial Officer. Whatever you do, you need to align with their procedures. As far as a name, whatever the church and your group mutually find workable is fine.
The only thing you cannot call yourself is a PTA (Parent Teacher Association) as that is the name of a very specific organization. PTO is generic and there are various other names parent groups give themselves.
Yes, a parent group (especially affiliated with a church or private school) can be part of the school/church rather than it's own entity.
In public schools, it's more common for the PTO to be an independent entity. While it certainly coordinates closely with the school, the group is responsible for its own actions, manages its own funds. The school is not "responsible" for this group.
If you stay under the church's umbrella, then you are basically just a committee like any other church committee. The funds belong to the church, although certainly they can be earmarked for a specific purpose much like Building & Grounds or a Mission group.
In your case, the church dictates how funds are handled. You need to talk to them about if you can have a separate bank account, what documentation to use.
We are a small preschool inside of a Presbyterian church, owned by the church. We don't have our own license and just run as a ministry of the church, and are a 501(c)(3) bc the church is.
I was just hired as the director and I know I cannot run the school and be the fundraiser, too. I want fundraising to be in the hands of a parent group who operate in an informal way to raise/spend money.
Does a PTO (we don't have to call it that) HAVE to incorporate or become its own 501(c)(3) with an EIN?
Can it just be a bank subaccount of the church?
Or can I go to a bank, and just open an account for the informal group to use for funds?
We are just looking for this: The easiest way to go set up an account (under church, or separate) for fundraising/booster money to go IN and OUT. We don't want to mess with incorporation/EINs....do you have to???