Jerusalem is a safe city and as a tourist visiting tourist sites you will have no problems. However you must follow the news in case there is a flare-up of violence in certain neighbourhoods. Check with locals before you set off.

The Dead Sea is several hundred meters below sea level and much warmer than Jerusalem. It seldom rains there. In any case you don't "swim" in the Dead Sea, you just float on your back for 10 minutes for the unusual experience. The majority of the day will be spent visiting Massada and the Ein Gedi Nature Reserve.