Definitely recommend the tram & various hikes on Mt San Jacinto. The hike to the peak is moderately strenuous but worth it. It can be quite chilly & windy at the top (& sometimes rainy), so don't go without appropriate clothes. If the peak isn't your thing, there are also several other nice trails to check out near the tram. Be sure to smell the Jeffrey Pines; they smell like butterscotch!
I've done some Joshua Tree hikes in the heat, and I wouldn't personally do any but relatively tame hikes in the summer. Bring at least 4x the water you think need!
Idyllwild is a little bit of a drive but very charming, and also will have more favorable temps due to the elevation.
Even with the heat, the Marriott pools will be nice, and they have misters as well to help you cool down.