"The Church" is the Catholic Church and all it's various Rites. (Rites are mostly geographic divisions of the Church, each with it's own customs and practices. Some Catholic Rites even have married priests, and so forth. In today's globalized society it has less to do with geography - there are Scottish Rite Catholic Chruches and Greek Orthodox Catholic churches all over the place.)

The Pope determines what the Catholic Church does and does not consider official teachings.

Christ died and was buried, and three days later rose in fulfillment of the scripture, ascended into Heaven, and is seated at the right hand of the Father. That's a teaching of the church.

Mary, mother of God, also ascended into heaven. That's a teaching of the church too, but until 1950 it wasn't "official".

Noah took two of every animal on his ark. That's a story in the bible, and there are valuable lessons in the story about God and his relationship with people. But if a Catholic priest tells his congregation that this is allegory, he's not going to be excommunicated because the church has no official position on that.

Your question about the value of papal infallibility indicates a misunderstanding of the term. A Pope can exercise Papal Infallibility by declaring a writing of his to be infallible. If Pope Benedict wanted to, he could declare infallible a writing of his saying that birds can't fly. Anybody saying that they've seen birds fly would be excommunicated until they changed their mind. However, he wouldn't do that because it would tear the church apart.

The right of infallibility is rarely exercised because of the effect it has on the church when it is.

Infallibility's value is the same as the value of the ability of legislature to pass laws. The Pope (and various ecumenical councils) are the source of the church's laws.