[Galaxy Note 3 General] jisoo:
As far as I know, all Note 3 versions in Korea are using the Qualcomm Snapdragon 800 chip and support LTE-a. However, this is really just normal LTE with Carrier Aggregation (CA) support (using two LTE bands simultaneously) added and is fully compatible with "normal" LTE.

KT does not do real CA as they have two bands right next to each other (15+20 MHz). In addition, even SKT and LGU+ have two bands available only in limited areas, so most of the time you would be using single band LTE anyway. Also, older devices (S2, S3, Note 2 and the first version of the S4) do not support CA at all but have no problems with the networks, so you really don't need to worry about "LTE-a" -- it's something the handset and the LTE basestation would negotiate and would not be used unless both ends are willing to use it.

Regarding the phone you have, I would guess that the internal hardware is probably identical to the SM-N900S (SK Telecom version) and only the outer case is different. I've seen some similar stuff at the Samsung R&D labs in Korea, where they use pre-release or internal models for testing purposes (as they have little retail value the engineers can test with these without any budget costs).

However, there's no guarantee all the hardware and partition layout is identical to the commercial models. You can try flashing a recovery (CWM or TWRP) through Odin to back up your system before you try another ROM. If the phone is on 4.3 JB, you should be able to use any of the 4.3 JB recoveries for N9005.

Keep in mind that if you flash any 4.4 KK stock ROM (even the modified ones), these will overwrite your bootloader. This is irreversible and may permanently brick your device.

4.4 KK cyanogenmod-based ROMs are safe(r) because they do not touch the bootloader. If you have any problems with these, you can always restore your 4.3 stock ROM you backed up from CWM or TWRP.

I'm not sure if you are talking about CWM (clockworkmod recovery) or CM (cyanogenmod ROM), but neither will work.

For CWM recovery, you need to use this one: http://d-h.st/rZq. It's made for the hlteskt version but works on hltektt and apparently on the hltelgt.

For CM, we are very close to having a working version on the 4.4 KK bootloader. I haven't been able to confirm if we need a different kernel, but at this moment it seems that at least the initial ramdisk (included in the boot.img) needs to be packed with a different offset, and the install script (updater-script) needs to be modified to support our Korean models. I should have an update on this later today.

I'm sorry if that was too technical, but I wanted to make the situation absolutely clear.

For the users on the LGU+ models, there is an additional catch. If you are in Korea and on the LGU+ network,

you will not be able to do voice calls on anything except the stock LGU+ stock ROM

. This is due to the way LGU+ has decided to route voice calls over the LTE network (this is not VoLTE but something much uglier), and users on any recent devices using the LGU+ network will be stuck on stock ROMs unless support for this weird network is built into cyanogenmod (which looks extremely unlikely because there are no devs working on this).

LGU+ models (hltelgt) should work OK on other networks, but you do need to wait for the KK bootloader support updates from the Korean devs before there's anything you can try out.