You should be able to walk in the store and ask for a new one.
As for enabling LTE. You don't need to flash any rom, stock is fine.
You need to:
-unlock the bootloader using fastboot (check the guide threads in the root section)
-flash custom recovery like
TWRP using fastboot
-download LTE enable package (try
this or
this), flash it using TWRP
-preferably factory reset your device using TWRP before rebooting, this make sure the 4G LTE settings sticks
When flashing the LTE enable package, it will add these three lines to the build.prop:
telephony.lteOnGsmDevice=1
ro.telephony.default_network=10
ro.ril.def.preferred.network=10
The preferred network set to 10 means Global mode, it allow the Nexus to automatically switch between 2G/3G/4G. The Nexus do not have hardware technology to make or receive called while connected to LTE. This Global mode will allow the phone automatically revert back to 3G when making or receiving calls. The phone can't do this if its set to an incorrect mode.
These build.prop changes only take effect after a factory reset. You can instead modified settings while in Android OS to change these values without modifying the build.prop, but you may run into issues where after rebooting the 4G settings doesn't stick and the phone revert to 3G, which is why its better to put the changes into the build.prop.
Afterward, for security, you can reflash stock recovery using fastboot, then relock the bootloader. Or leave it the way it is if you want to flash other things. However, unlocking the bootloader again require the phone to be completely wiped.
One thing to note, Google OTA updates will not work with a modified build.prop and radio, which is what the LTE enable package modified. So applying OTA updates will cause the phone to soft brick. To prevent this you need to restore the phone to stock before applying the update, you can use TWRP to make a backup image of the phone in its original state before flashing the LTE enable package.
What I would do after installing TWRP is:
-Create a backup image of the phone in its stock form, uncheck the "cache" and "data" partition from backing up so it won't save your user data.
-When you receive Google OTA update, boot back into TWRP, create another back up image.
-Use TWRP to restore your original stock backup, since you did not back up "data" partition for it, it will not overwrite your present apps and settings.
-Now there are two choices, you can either use fastboot to flash back the stock recovery.img, then boot into the phone and run the OTA update within the phone. Or if you downloaded that OTA update package in zip format, you can boot into TWRP and flash that OTA zip, provided that the TWRP version you are using at that moment is compatible with that latest OTA zip.
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