The answer to the first question is positive: yes, you can safely visit Hebron by night. (Old Hebron, that is - the area right around the Cave of Patriarchs/Ibrahimi mosque).
As for the second question... well... Hebron is not exactly a party-oriented city. Other cities in the region are much better when it comes to nightlife. Tel Aviv or Ramallah both have fun restaurants and cafes to chill out and relax. Hebron is definitely a lot more conservative. So, forget about partying like crazy...
I can only think of one reason to come to the city by night: its centerpiece. Seeing the Cave of Patriarchs & Matriarchs /Ibrahimi mosque when its dark all around, is actually a pretty nice experience.
The 2 minarets of the Cave/Mosque are illuminated in green, like other islamic holy places in the area.
The Synagogue/Mosque at night.
There might be very few visitors around; or even, just you. But the site is guarded 24/7, so you are pretty much safe, as long as you stay in the main area.
Entrance to the Cave itself is usually possible during part of the night. For example, on a normal day, the synagogue is open from 4 AM until 9 PM in the winter. (10 PM in the summer). It is free of charge.
The easiest way to go to Hebron when it's already dark is to take the Egged bus 160 from the central bus station in West Jerusalem (tahana merkazit, in Hebrew).
Get off at the first stop in Old Hebron: you will be very close to the Cave. The drivers usually announce this stop, but let him know in advance that you are going there, and that you want to get off there.
Buses run between Jerusalem and Hebron until about 11 PM. However, there are no buses at all between Friday afternoons and Saturday nights, due to the jewish Shabbat. Check the daily schedules on the Egged website: www.egged.co.il/Eng/ . Also, you can ask at the information desk of the bus station in Jerusalem for the schedules: they will print them for you, for your round trip, and for the entire week.
Get off at the first stop in Old Hebron: you will be very close to the Cave. The drivers usually announce this stop, but let him know in advance that you are going there, and that you want to get off there.
Buses run between Jerusalem and Hebron until about 11 PM. However, there are no buses at all between Friday afternoons and Saturday nights, due to the jewish Shabbat. Check the daily schedules on the Egged website: www.egged.co.il/Eng/ . Also, you can ask at the information desk of the bus station in Jerusalem for the schedules: they will print them for you, for your round trip, and for the entire week.