Flash (firmware update) on 8 years old original Samsung Galaxy Tablet 10.1 GT-N8013 with latest Android 11 is possible now. theupgradeguide.com provided detailed steps and screenshots nicely to follow.
html6405 from xda-developers provides all the Downloads for : Samsung Galaxy Note 10.1 Wifi. Make sure that the latest TWRP recovery file and magisk_patched_n8010.img file are download from same place, instead of img file from twrp official site. This also applies to LineageOS and Open GApps zip files. Official sites might not provide support for GT-N8013 anymore.
TWRP image replace recovery.img file on GT-N8013, which is done by Odin or Heimdall (Jodin is Java wrapper around Heimdall) via Odin mode, aka bootloader mode. TWRP img is used in recovery mode: power + volume-up (release buttons after Samsung screen is displayed; volume-up key is on the left to SD card slot)
MagiskSU from topjohnwu replaces SuperSU for rooting recent version of Android devices. Magisk Manage app is included in Lineage OS, which will be used for creating magisk_patched_*****.img in place, so you can rename it as boot.img and flash boot partition in Odin mode (or just select "Direct Install" method, which will flash boot image in place.). After reboot, GT-N8013 is rooted, and passes SafetyNet Attestation checking via Magisk.
LineageOS 18.1 on GT-N8013 from htm6405: lineage-18.1-20210114-HTML6405 seems to have the common issue on LineageOS with charging over a hour, which might freeze and could not resume from standby screen. Hopefully this will be resolved by next release.
LineageOS boot up process is a little slow (initial boot is even slower), compared to original Samsung boot. I did not find an option to fast boot it yet.
After flashing GT-N8013 with LineageOS 18.1, Android display device info as GT-N8010, I understand the img file I used did have subfix with n8010. It is fine.
Android (Trust System security) complains that This build was signed with public key, and SELinus is not enforcing, you system has been weakened, so I had to disable the trust, not the best practice really.
Also credit to Sonny Kurniawan on Blogger, who provided details on flashing Android Nougat. Yes, my stocked GT-N8013 has been collecting dust all these years. 😅
Always xda link for more and latest info.
As to OEM bootloader unlock, JJEgan from xda-developers says GT-N8013 is unlocked. I enabled Developer debug mode in original ROM (Android 4.1.2), and I don't recall there is option for bootloader unlock either.
I used Heimdall on OS X Mojave to flash TWRP and magisk patched.img.
Update: I flashed GT-N8013 with LineageOS 18.1 BETA from html6045 with same version of TWRP as ALPHA release last December without any issue.
Note: Magisk is not installed on BETA release.
Tip: LineageOS might not include some language input method, but Gboard app from Google Playstore is pretty good, which detects extra language selected, and download most popular input method in the background.