How to pronounce '蒙古' correctly?
The Pinyin for 蒙古 is written like this in ABC:

Mẹ̌nggǔ

Notice the dot under the letter E, this is an indicator that this word is read with tone sandhi. The tone sandhi rules dictate that meng3 gu3 be read:

meng2 gu3 (méng gǔ)


Another factor that may add to your confusion is the fact that the standard pronunciation in Taiwanese Mandarin is different. For instance, if you check MoE's entry for 蒙古族 the pinyin for 蒙 is clearly written as:

méng

蒙 is never read měng in TW.