All three are close to the park entrance.
Where to stay in Zhangjiajie
Sleep in Wulingyuan, the town at the gate of the national park. You'll spend most of your days in the park, so being close saves you hours.
The hotels I'd pick
How long to stay
Four nights. Give the park at least three days, every viewpoint is different. Here's what I'd do, in order.
Getting there
From Furong Town it's 23 minutes by train to Zhangjiajie West, then about 30 minutes by taxi to Wulingyuan. The same train goes both ways.
Quick answers
Should I stay in Zhangjiajie city or Wulingyuan?
Wulingyuan, the town at the park gate. You'll be in the park most days.
How many nights do you need in Zhangjiajie?
Four nights, with at least three days in the park.
How far is Tianmen Mountain from Wulingyuan?
About 40 minutes.