Can I somehow unlock my phone's bootloader or root it? My Moto X 1st edition (XT-1060, career Verizon, OS 5.1) doesn't stay rooted. I mean to say that it turns back to unrooted state wherever it reboots or even when it is left untouched for an hour or so. Any system apps that I've uninstalled while rooted also shown in super user applist and any other root related changes e.g changed font also reverts back..

I've tried many tools to preform root operation like iRoot but none of them does the job. Kingo Root however does the job but as I mentioned the phone reverts back to unrooted state after being rooted.

Moreover, all the tutorials and guides about rooting my phone mentions in the first place that I need to unlock the bootloader first. I have tried the official method for unlocking the bootloader but it simply does not work. Just to mention, official method requires following of some steps after which phone shows some strings and you can paste those strings on Motorola's site to know if your phone has an unlockable bootloader or not. They say my phone's bootloader cannot be unlocked. It is also not the developer edition of Moto X otherwise the things were much easier. It is a carrier specific version  of Moto X i.e. Verizon and the official Moto website says about such phones that:

Please keep in mind there are many factors as to why a device is or is not unlockable. Carrier exclusive models are almost never unlockable

So my question is that:

Can I somehow unlock my phone's bootloader or root it? 

My actual concern is about rooting it so if it can be done somehow without unlocking the bootloader, I'll be more than happy. I included bootloader unlocking as a part of my question just because I cannot see any way without doing so.
I have the developer edition of Ghost, so never had this problem with unlocking the bootloader.  Did you try RockMyMoto?  You probably have the camera patch by this time, so this might be the method you need.

If that works, I suggest you flash one of the 7.1.2 ROMs over there at XDA after you get TWRP 3.1.1.  Mine is rocking Xperience right now.  Your phone will thank you :)