To what extent must one be cautious about swallowing blood while fasting, and how should this caution be exercised? Is it obligatory to remove blood stuck between the teeth, and how can one deal with excessive spitting resulting from the obsessive thought that blood is present?

My advice to the one afflicted by waswas (obsessive whispering/doubts) is to turn away from these waswas and ignore them.

He should not spit excessively, nor should he conclude that blood has exited his gums merely based on doubt and delusion. Rather, he should not do any of that unless he is absolutely certain that blood has exited. Only then should he spit it out.

With doubt, nothing is incumbent upon him. He should not use a stick to verify, as that increases waswas and may cause bleeding. Rather, if he tastes blood in his mouth, he should spit it out; otherwise, he should conclude that nothing has exited.