I teach at a C of E school.
Selection based on faith is waaaay down the list, but is there. The majority of our children are not christened, don't go to church etc. We are just the local primary for them. We have children of many faiths too. And a small % who have travelled out of catchment that come because we are a church school. None of our parents have ever requested their child be removed from worship but then they come knowing we are a church school.
We are run slightly differently. The vast majority of our funds come from the local authority as usual. Occasionally, the church will fund part of a project, like our extension. Which was funded 5% by the diocese. Our school does not fund the church in any way.
A member of the diocese was on the selection panel for our headteacher's interview and the headteacher had more leeway over appointments until recently.
We have a different curriculum for RE. It's still broadly Christian in nature, probably more Christian than your average primary school. We are inspected separately for RE. All schools have to have a collective worship of broadly Christian theme, ours probably leans more towards Christianity than a non church school might, but is not only Christian in nature.
We sing traditional hymns and more modern songs. As does my child's non-church primary.
We have assemblies daily, including the vicar coming once a week. As does my child's non-church primary.
We've been more and less 'churchy' depending on our SMT. At one point (a long time ago) our headteacher was a vicar.