Samsung GALAXY Note3
The latest firmware for Samsung GALAXY Note3 (SCL22) is SCL22KDS2GPL1 with Android 5.0, released on December 21, 2016 for 1 region (CSC). Every official firmware build for SCL22, newest first: one row per build with all regions (CSC) that received it.
Each firmware build has its own page at https://samfirm.shop/build/{PDA}, where {PDA} is the build number from the list above. Example: https://samfirm.shop/build/SCL22KDS2GPL1
SCL22 started on Android 4 and has received 1 major Android upgrade; it now runs Android 5.0. The newest build, SCL22KDS2GPL1, was released on December 21, 2016. No new build has appeared for 119 months, so regular updates for this model have most likely ended.

SCL24

SCL23
The newest firmware for SCL22 runs Android 5.0: build SCL22KDS2GPL1, released on December 21, 2016.
The latest build for SCL22, SCL22KDS2GPL1, was released on December 21, 2016 (9 years ago).
SCL22 has had no new firmware for 119 months, so regular updates have most likely ended.
Yes, all firmware files for SCL22 are 100% official, original, and obtained directly from Samsung servers.
To flash SCL22, you need to download the appropriate firmware, put your device into Download Mode, and use the Odin tool to load the BL, AP, CP, and CSC files.
Downgrading is only possible if the bootloader version (binary bit) of the old firmware matches the current one. Always check the binary version before flashing SCL22.