Cooking on Shabbos is forbidden form the Torah even with derivatives of fire. Even slightly roasting an egg by placing it next to a kettle is assur from the Torah.

Cooking in the sun itself on Shabbos is totally permissible. Cooking in תולדות החמה- like a hot scarf heated by the sun- is assur rabbinically so it’s not consumes with תולדות האור. But r Yosi permits it and makes no decree

Even r Yosi agrees one can’t bury a raw egg is hot sand on Shabbos bc it either may lead to insulating with ash or muketzeh issues with moving the sand.

There was a story with ppl of Teveriah who set up b4 Shabbos a system of cold pipes running through the hot springs. The cold water would be heated automatically on Shabbos. The rabbis were against risk practice and did not allow the people of teveriah to use the heated water. The is either proof that fully insulating before Shabbos with material that adds heat is forbidden and/or that cooking on Shabbos with תולדות החמה is forbidden. R Yosi counters than the springs of teveriah are really תולדות האור bc they’re heated from fire of Gehonim.

On Shabbos, the Rabbis do not allow bathing even with water heated before Shabbos. However, one could use water heated before Shabbos to wash hands face and feet.

On Yom tov, beis Shammai does not allow one to heat up water to wash hands face and feet. Beis Hillel allows it bc of ״מתוך״- and this is a universal need.

Showering on Shabbos- r Meir forbids it even in cold water. R shimon allows it even in hot water. R yehuda forbids it in hot water heated even before Shabbos but allows it in cold water. The Halacha is like r yehuda