Is a person who experiences daily bleeding in the mouth during Ramadan exempt from rinsing the mouth, and is it sufficient for him to spit out the blood along with being excused from the traces of blood? And is he required to make up the fasts if he swallows saliva mixed with blood to avoid hardship? If he is not required to make them up, should he make up the days he completed fasting with the intention of making them up, believing that he had broken his fast?

If the bleeding is daily in Ramadan (definition), spitting out the blood is sufficient for you, and gargling is only obligatory for prayer or reciting the Quran. Swallowing some blood unintentionally does not invalidate the fast. Some scholars hold that for one who experiences frequent bleeding from the mouth, they are excused, and their fast is not invalidated even if they swallow it intentionally, due to the difficulty of guarding against it. As for the intention to break the fast, if you reject the intention to fast or actually break the fast with an invalidator, then your fast is invalid and you must make it up. However, mere hesitation or believing the fast to be invalid does not invalidate it.