With water heated up before Shabbos- one can’t wash their whole body but can wash hands face and feet. Rav holds limb by limb is fine (and maybe even the entire body except for one limb) and shmuel holds it is forbidden. The gemara seems to favor Shmeuls opinion that it is forbidden.

It used to be fine to bathe on Shabbos with water heated before, but the bath attendants were heating water on Shabbos so the rabbis forbade bathing on Shabbos. They still permitted sweating in the bathhouse, but people were really bathing and claiming they were sweating, so that was outlawed as well. It is however ok to bathe in the hot springs of Teveriah on Shabbos.

Someone who doesn’t listen to the Rabanan is called a “sinner.”

Included in the prohibition to bathe would be washing with cold water and then standing in front of a fire.

One cannot put pitcher of water close by a fire to reach יד סולדת הו, but it is ok for it jsut to take off the chill.

Regarding oil- r yehuda certainly holds that it’s exactly like water and can’t be by fire to reach יד סולדת but could be jsut close enough to take off the chill. R shimon Ben gamliel holds that by oil even jsut warming slightly is considered cooking and it’s forbidden unless one does it irregularly by smearing oil on the body and then standing by the fire. There is the tanna whose opinion is unclear. In one possibility, he holds oil is not subject to cooking at all even if it reaches יד סולדת בו

Water in a כלי ראשון has capacity to cook but in a כלי שני it no longer has capacity to cook.

One can’t talk torah in a bathhouse even in secular language. The one permit is to stop someone from doing a sin it is ok.

One can’t swim on Shabbos bc they may make a life preserver. It is questionable whether a large tub that has a run over it is included in this prohibition.