How to root Moto X with stock 4.4 kitkat without unlocking bootloader? I have tried RockMyMoto, PwnMyMoto and SlapMyMoto. All have failed. I am also unable to downgrade from 4.4 kitkat (which the phone came with) to 4.2.2 in order to root. The downgrade fails due to "security downgrade" or something similar. Any advice? I do not with to void the warranty by unlocking the bootloader.

Is it possible to flash a pre-rooted kitkat system image using RSD Lite? I was able to use it to re-flash 4.4 after a failed attempt to flash 4.2.2

This is somewhat urgent as it is a surprise gift for someone who likes having rooted phones.

Note: I only mention downgrading from kitkat because there seems to be no way to root a phone which came with kitkat on it. If there is another option, I would love to hear about it!

UPDATE: It is possible that the phone came with 4.4.2 and the updated bootloader. I did not check before I attempted rooting and flashed 4.4 using RSD Lite.
You have two questions here, which should be split into different questions.

To answer your first question from the title: In general, you need to unlock the bootloader to root the device.

Also, by unlocking the bootloader, you are allowed to flash a custom recovery. Using this custom recovery, you can wipe the device's cache, system, and data folders. Afterwards, you can flash a custom ROM that may have a previous version of Android.

Also, under stock ROM, (in some cases) some bootloaders (esp newer ones) will not allow you to downgrade. Doing so will break stuff on your device, like Wi-Fi and sound not working properly, etc. But again, this is a general statement (not specific to Moto X), and may partially answer your second question.

Anyway, there is a guide here that allows you to downgrade from KitKat to JellyBean.