[Galaxy Note 3 Q&A, Help & Troubleshooting] PhotonIce:
Alright, now since you have mentioned that you want to use lineage os, here's a guide for that too.

So currently, it is recommended to use android 11 lineage os nightly builds as they are stable and official, so they should not have any bugs. So unfortunately, I cannot recommend installing lineage os android 10, as you had originally wanted, though the installation process for that will be somewhat similar. It is still not recommended to use old or unofficial builds due to bugs and lack of security updates.

First, we need to download lineage os for this device onto our pc, here's the link to the latest nightly build:

https://mirrorbits.lineageos.org/full/hlte/20220319/lineage-18.1-20220319-nightly-hlte-signed.zip

Next we need to download a GApps package for google play functionality. OpenGApps is recommended as it is open source, secure and reliable, usually with no bugs. I recommend using the nano package instead of pico if your device can handle it (your's can handle it just fine), as it contains google play functionality, and some extra google features which cannot be downloaded from google play. Here is the link to the nano package for your device (only for android 11 roms):

https://sourceforge.net/projects/opengapps/files/arm/20220215/open_gapps-arm-11.0-nano-20220215.zip/download?use_mirror=udomain&r=&use_mirror=autoselect

Now, if you want to get your device play protect certified, you will also need to download magisk for rooting, and magisk modules (so that you can download netflix and use banking apps etc.), but that is something you can do later on, and you can find any tutorial online for this.

Once the download is complete, move the lineage and opengapps .zip files to an sd card and insert it into the phone. Enter TWRP, go to wipe and click on Advanced. From here, wipe System+Data+Cache+Dalvik/ART Cache. Your files such as photos and media will not be wiped.

Next, go back to the TWRP main menu and select install. Click the select storage button at the bottom left and click on Micro SD Card. Now, click on the lineage os zip file, then click on the Add Zips to Queue button and select the opengapps zip file, then click install. This will take some time, do not interrupt it. Once it is complete, you can reboot the phone. The first reboot will take some time, and the first boot may be slow, after setting up your device, make sure to reboot it at least once.

Congrats, you've just installed a custom rom with google play functionality via TWRP!

Unfortunately, sometimes there are issues. If you still cannot get into lineage os despite waiting at least 10 minutes on the boot screen, then follow these steps below:

The lineage team says that booting issues may be due to incompatible recoveries, this means that we will have to use lineageos's own recovery to flash the rom.

For this, download the lineage os recovery.img from here:

https://mirrorbits.lineageos.org/recovery/hlte/20220319/lineage-18.1-20220319-recovery-hlte.img

Now copy this image file to your sd card and insert it into the phone.

Enter TWRP. First of all, click on wipe, then click on advanced and select System+Data+Cache+Dalvik/ART Cache and wipe. Now, go back to the main menu and click install. At the bottom right or left, there will be a button to "Install Image". Click on that button, and select the lineage recovery zip file and install it. Now go back to the main menu and click on the reboot button, then click on reboot to recovery.

Now you will boot into the lineage os recovery, from here you can install lineage os and the gapps zip files. Here is a tutorial:

This tutorial is for a oneplus phone, but it will be the same process on your samsung device.

Once it is complete, you can restart the device. If you would like TWRP back after flashing, just follow the steps from my first post and use odin to flash the twrp file again. I recommend that after installing lineage os, flash twrp again, as twrp recovery is much better than lineage recovery, and you dont need lineage recovery after installing lineage os.