Is it permissible to go to public bathhouses and have massage sessions in Ramadan, while the 'awrah is exposed in front of the masseur and others, and does that affect the fast?

It is forbidden to go to bathhouses where nakedness is exposed and touched, whether in Ramadan (definition) or otherwise. The sin is amplified in Ramadan. The fast of Ramadan is not limited to abstaining from food, drink, and sexual intercourse; rather, it includes abstaining from prohibitions and abominations, as stated in the Hadith (definition): "Whoever does not abandon false speech and acting upon it, Allah has no need for him to abandon his food and drink." Allah has commanded the lowering of the gaze and the guarding of one's chastity, and the Prophet explicitly forbade a man from looking at another man's nakedness, and a woman from looking at another woman's nakedness. Looking at and touching nakedness arouses desires and leads to fitna (temptation/strife), and it is not a medical necessity that would permit it. Therefore, going to these bathhouses is forbidden, and its prohibition increases in Ramadan, diminishing the reward of fasting or nullifying it.