Is it permissible for a fasting person to swallow something smaller than a fava bean, and does that break the fast? Is it permissible to swallow residual water after rinsing the mouth and spitting? And what is the ruling on water entering a child's (i.e. penis') opening, and does that invalidate the fast?

It is not permissible for a fasting person to intentionally swallow anything, even if it is smaller than a fava bean. If one does so intentionally, their fast is broken, and they must make up the day and repent. One must be careful about food remnants in the teeth. If it was possible to remove them, but one did not, and swallowed something, then their fast is broken, and they must make up the day. This is the view of the majority of scholars. However, if something enters unintentionally, there is no blame on them. It is not permissible to swallow residual water after rinsing the mouth. However, swallowing saliva after rinsing the mouth and expelling the water does not break the fast. As for exaggerating in washing the private parts to this degree, it is blameworthy and a form of extremism in religion. A woman is not required to wash the inside of her private parts, but only what is visible when relieving herself. If what was mentioned occurred during fasting, making up the day is more cautious.