Is it permissible for a person who cannot stand (disabled/paraplegic) to lead congregational prayers if he has memorized parts of the Qur’an? And does swallowing a liquid that comes out of the stomach without vomiting affect the fast? And if it comes out during prayer, does it affect it?

If a fasting person swallows a tiny bit of leftover food stuck between their teeth, it does not affect the validity of their fast or prayer. As for swallowing vomit during prayer, if it comes out involuntarily, the prayer is not invalidated unless the worshiper intentionally swallows it back after it has come out. If one intentionally vomits, their prayer is invalidated. Similarly, it is invalidated if the vomit is impure and has changed from the state of food.