Does Google Nexus 5 has arm or armX86 or armX64 processor? I am trying to install Xposed module on my rooted Nexus 5 running stock  Android 5.1.1. In that process I am trying to download the files from [this XDA page. I am unable to find whether my Nexus 5 is an arm or armX86 or armX64.

Sure, I can try one after the other as I have a nandroid backup, and I can also Google it and find it in pages like this.  But, how can I be sure of myself? I can rely on any app that might say so after installing in my phone or some reputed sites say it so. What would be the best and authentic way for me to find the same?
The Nexus 5 runs a Snapdragon 800, which is not a 64-bit supported processor. That means you should be going for the arm version of Xposed, and not the arm64 version.

NOTE: You are getting confused with the meaning of x86. In this context, x86 refers to mobile processors made by Intel (the Nexus 5 has an "arm" chip). The difference between "arm" and "arm64" is whether it supports 64-bit execution. There is no such thing as "arm86".  

Currently, there are only unofficial builds of Xposed for 5.1 and higher.