Is it possible to permit showering on the Sabbath: 1. with water heated by the solar collectors of the water heater before the Sabbath and also during the Sabbath; 2. water heated by an electric boiler before the Sabbath, and also water heated by a boiler through a timer that operated on the Sabbath?

I think this is more connected to an unintended act in a case of an inevitable result. That is why they prohibit it.

From a conversation I once had with a plumber, I understood that the mixing of the hot and cold water is done indirectly. According to the Beit Yosef, there is no problem washing one’s face, hands, and feet on the Sabbath even with hot water that was heated on the Sabbath, provided it is water heated by the sun, because in that case they are lenient. There are disputes regarding the pipes—whether they are considered something heated by the sun’s derivative heat—but in any case there are also those who permit that. As for the boiler itself, whether during the Sabbath or before the Sabbath, that would only be if we say that it is indirect causation. The whole prohibition on washing the entire body is because of the bathhouse attendants / bathhouse decree, lest one stir the coals. Rabbi Aviner says that the law of the bathhouse decree does not apply to a particularly delicate person, and all of us who shower every day or almost every day are considered such. By the way, there is no prohibition on immersion in hot water; that is not included in the bathhouse decree. Rabbi Aviner says that even for an Ashkenazi, if this harms his Sabbath enjoyment, he has authorities to rely on. Regarding limb by limb and not the whole body, Rabbi Shlomo Zalman Auerbach writes that the shower head commonly found in showers, which releases water through its holes, is considered limb by limb and not the whole body. According to Sephardic practice, bar soap is permitted, and it is not like melting ice into water, so it is not creating something new. I also heard that there is an opinion that squeezing does not apply to hair.