Is it permissible to build tourist hotels on the shores of the Dead Sea, and is it permissible for an engineer to work on implementing air conditioning systems in a hotel that includes facilities serving alcohol and mixed swimming pools in this location?

If tourist places are often used for disobedience to Allah, then it is not permissible to participate in their construction or repair, or to install anything in them, as this constitutes aiding in sin.

As for the Dead Sea area, there is no شرعي (Shar'i) evidence that it is an area of torment. The default rule for things is permissibility. Therefore, it is permissible to build near it, benefit from what comes out of it, such as salt, and bathe in it, provided that it is free from forbidden mixing and nudity. The mere mention in books of Tafsir (exegesis) that it is the area of the people of Lut does not have an effect unless it is supported by an authentic report from the Infallible (Prophet Muhammad), peace be upon him.