Please advise me about how to travel Mt. Huashan between China national week (between 3 - 7 oct)

Well, it is possible to travel to the mountain and back to the city in one day. You can catch a high speed train from North Railway Station to Huashan North Railway Station. The earliest one departs at 06:23 and the duration is around 40 minutes. Upon arrival, you can take a taxi to the scenic spot directly. When you finish visiting, you can also take a train back, and the last one departs at 20:27.
If possible, you are advised to reach the scenic spot in the afternoon of the one day before. Then you can climb the mountain in the late afternoon. When you climb on the top of the mountain, you can have a rest, and then enjoy the sunrise on the next morning.
Tip: You can rent the down jackets at the top of the mountain, since it will be very cold at night.