Can anyone please tell me if Tianmen Mountain and Tianzi Mountain Nature Reserve are the same?

No, they are not the same area. Tianzi Mountain is located inside Wulingyuan scenic area, and the Tianmen mountain is a solitary scenic spot located at Yongding District of the city, and the distance is about 35 miles between them.

Hi look up the book "Now Anyone Can Explore China", it will provide you with all the information you need to do a self guided trip. On Tianmenshan you don't need a guide however you will need a LOCAL guide on, Chinese Name: 天子山 (Tian Zi Shan). I can personally reccommend Chris Tan as a very good local guide.

TianmenShan has the glass plank road, it is nearby ZJJ

TianziShan is nearby Wulingyuan where there are good places to stay.
BTW, Shan simply means mountain. There is heaps to see especially on TianziShan so I suggest you stay on the mountain at least 4 days. Wulingyoun has heaps to see too especially the World's longest glass bridge over the grand canyon.

ZJJ has an amazing art gallery, it is called "The Sand Stone Art Gallery", it is not to be missed.
Make sure you see the local culture shows too, they are totally stunning!!!

A word of warning, don't go near the place in May, it is Chinese holiday time and just teaming with visitors.

You must also spend 3 or four days in Phoenix Ancient Town (Fenghuang). You can take a bus there from ZJJ. Accommodation is not a problem.

Enjoy your trip!