1) Yes you can leave luggage with bellman at your Xi'an hotel. Alternate is Left Luggage at the train station.

2) Watch the forecast and pick a non-rainy day. Seriously. Abandon the plan if it's raining.

3) Huashan area is not just one mountain. Do your internet research on the different peaks and paths...the info is out there. Also on the (mostly crappy) places to shelter for the night on the mountain. Parts of some trails/paths are very dangerous and that includes the infamous Plank Walk--surprised that thing hasn't been closed down yeet. Be very careful, this mountain has more than its share of seriously injuries and fatalities, compared to the others around China.