A 100 percent flashable ZIP file based on the stock ROM is now available for download courtesy of XDA developers. It is deodexed and can be rooted simply by flashing superSU-2.02 zip from chainfire. There is simply no other way to attain root on it.
There is also a deodexed + rooted build available which also has unsecure boot. Also if you own a Galaxy S5 mini, you can find its deodexed firmware here.
Here are the download links for rooted and deodexed stock ROM for Samsung Galaxy Tab 3 10.1:
100% stock: STOCK-4.4.2-P5210(fixed).zip (100% stock, it is fully bloated, odexed, and non-rooted).
Deodexed + rooted: STOCK-ROOTED-DEODEXED-4.4.2-P5210(fixed).zip
To flash it, download ODIN on to your computer and put the firmware in PDA field and flash it. Alternatively you can flash it in philz recovery.