Are food remnants and stomach secretions that rise to the mouth with a belch considered ritually impure (najis) like vomit? How does one purify themselves from them if they do not exit the mouth? And must a worshipper intend to recite Al-Fatiha in prayer?

Regurgitation is impure, like vomit. According to the Maliki school, it only becomes impure if its state has changed. So, what has changed is impure, and what has not changed is pure. Its purification is by washing the mouth if it reached it.

As for Al-Fatihah, the intention to recite it is not a condition, for the intention of the prayer encompasses all its parts, and the intention at the beginning of the prayer is sufficient.