[Galaxy Note 3 Q&A, Help & Troubleshooting] PhotonIce:
From you're post, it seems as though you are having issues with TWRP. Below I will give you a full tutorial on how to get TWRP up and running for your specific device. Do give this a go, as I have done the research for your specific model, and it should work just fine.

First things first, unlock your devices bootloader. You may not have done this correctly, so if it is possbile to boot into android, then try this:

How to unlock Bootloader on Samsung Galaxy Note 3 | Softstribe

Unlocking bootloader in Samsung Note 3 made easy and you can simply enable developers mode where you can perform OEM unlock.

softstribe.com

If you're bootloader is already unlocked, then proceed.

Now that the bootloader in unlocked, we need to install TWRP for your specific device. Your device is the N9005, which is the international snapdragon model. It's codename is hlte, or hlte*. For TWRP, it is also categorised as the Europe/American model (hltexx).

For flashing TWRP correctly to a samsung device, we will need odin. Odin is a flashing software for samsung devices.

Download Odin.zip from here:

Samsung Odin - Root Android download page for one click root

Samsung odin download - Root android and oneclickroot for flash / install custom ROM with one click root method

odindownload.com

(Download the latest version (v13.4.1 at the time of writing this)

Now, extract Odin.

Once we have Odin downloaded, we will need to install the correct usb drivers for Samsung phones.

Download the file below, then extract the zip and run the installer:

Samsung_USB_Driver_v1.7.31.0

MediaFire is a simple to use free service that lets you put all your photos, documents, music, and video in a single place so you can access them anywhere and share them everywhere.

www.mediafire.com

Now that odin and the drivers are ready, we need the actual twrp image which we need to flash to our device.

Here is the link, download the latest .img.tar file (3.6.1_9-0 at the time of writing):

Download TWRP for hlte

Download TWRP Open Recovery for hlte

dl.twrp.me

Do not download the normal .img file, as that cannot be flashed via odin.

Now that we're past the prerequisites, we need to actually flash twrp to your device. To begin, we must put your note 3 into download mode. To enter download mode, power off your phone completely and then press Power+Home+Vol Down at the same time and keep these three buttons pressed until your phone shows a "download mode, do not power off target!!!" screen, you can follow this video tutorial below:

Now that we are in download mode on the device, we need to get back to our computer and configure odin to flash twrp. First things first, open odin. Now, connect your phone to the computer. You should see a little "COM-" show up in one of the empty boxes in the top row under the word "ID:COM". This is good, it means that your pc successfully recognises your device.

Now, go to the options tab on the left side, makse sure that repartition is

UNCHECKED

. Now, click on the "AP" button, and navigate to and select the TWRP.img.tar file that you downloaded earlier. After you have selected it, click on start at the bottom right. Now it will begin to flash twrp, which should not take long. Make sure that you do not disconnect the device while flashing, wait for the phone to complete. Once it is done, the phone will reboot.

Now, just remove the battery, then put it back in. To enter TWRP recovery, keep Power+Home+Vol Up pressed, then once you see the samsung boot logo upon the phone powering on, let go of power, but keep home and volume up pressed.

Congrats! You've just flashed TWRP to your device, amazing!

Now you can go and look for custom roms to flash to your device. Since your model is the N9005, look for hlte/hlte* roms, don't use any roms if they don't support the hlte/hlte* variant.