The latest firmware for Samsung Galaxy Tab4 8.0 (SM-T337A) is T337AUCS2BOH5, released on January 23, 2017 for 1 region (CSC). Every official firmware build for SM-T337A, 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/T337AUCS2BOH5
Specifications of SM-T337, a regional variant of the same device. Chipset and SIM lines are left out: they can differ by region.

SM-T337V

SM-T330X

SM-T332

SM-T230X

SM-T320X
You can find the firmware on the model: SM-T337A page.
Yes, all firmware files for SM-T337A are 100% official, original, and obtained directly from Samsung servers.
To flash SM-T337A, 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-T337A.
Yes. Every build on this page is the official stock ROM for SM-T337A from Samsung's update server, the same package other sites call the flash file or the software update. It installs with Odin on a Windows PC.
No. Factory Reset Protection is tied to the Google account that was on the phone and survives a firmware flash. Sign in with that account or use the legal options in our FRP guide.