Samsung Galaxy Note 3 SM-N9006: Chinese knockoff/clone or genuine phone?
It seems that your device uses a MediaTek MT6582 chip, as seen in this line of your device info:

Baseband: MT6582


There are no Samsung Galaxy flagship devices (e.g. Note / S devices) that use MediaTek chips. Samsung uses Qualcomm Snapdragon chips, its own Exynos chips, and the occasional Spreadtrum chip. MediaTek chips just don't deliver the level of performance that Samsung wants in their flagships. I'm afraid the device is a clone.

One way to confirm this is by going to the dialer app and typing *#0*#. On a genuine Samsung device, you'll be taken to a debugging menu right after you enter the last #.

After some more research and consideration, it does appear that your device is indeed genuine. The *#0*# test you did (and passed) further strengthens this claim. (I apologize for my previous answer.)