Some churches formally seek recognition of exemption from the IRS, but they are not required to do so and the IRS does not have a comprehensive list.  A church can have 501(c)(3) status by virtue of its activities, whether or not it files with the IRS.  (See Ready Reference Page: “What Constitutes ‘Church’ Eligible for Exemption.”)

Whether you have the right to review the books and records of the church depends on the type of entity it is.  If it is a nonprofit membership corporation and you are a member, you probably have legal rights to review the books and records for a proper corporate purpose.  But many churches, probably most, are not organized in that form and the parishioners have no such rights.  In that case, your recourse is probably to find a more congenial church.