
Samsung Galaxy Note3
The latest firmware for Samsung Galaxy Note3 (SM-N9008V) is N9008VZMUDPA2 with Android 5.0 (Jelly), released on January 26, 2016 for 1 region (CSC). Every official firmware build for SM-N9008V, 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/N9008VZMUDPA2
SM-N9008V started on Android 4 and has received 1 major Android upgrade; it now runs Android 5.0 (Jelly). The newest build, N9008VZMUDPA2, was released on January 26, 2016. No new build has appeared for 130 months, so regular updates for this model have most likely ended.
Specifications of SM-N900, a regional variant of the same device. Chipset and SIM lines are left out: they can differ by region.

SM-N920AZW

SM-N920VZW

SM-N950UZK

SM-N920AZK

SM-N950X

SM-N930V

SM-N9300

SM-N930S
The newest firmware for SM-N9008V runs Android 5.0 (Jelly): build N9008VZMUDPA2, released on January 26, 2016.
The latest build for SM-N9008V, N9008VZMUDPA2, was released on January 26, 2016 (10 years ago).
SM-N9008V has had no new firmware for 130 months, so regular updates have most likely ended.
Yes, all firmware files for SM-N9008V are 100% official, original, and obtained directly from Samsung servers.
To flash SM-N9008V, 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 SM-N9008V.