Short answers, pulled from the story.
The Persian name Moghulistan translates directly to Land of the Moghuls. This term refers to the eastern branch of ethnic Mongol Chagatai Khans who ruled the region from the mid-14th century until the late 17th century.
Tughlugh Timur converted to Islam on the 2nd of May 1354 alongside 160,000 followers. This event ultimately marked the Islamization of Moghulistan and inspired his territorial expansionism into Transoxiana.
Moghulistan in its strictest sense was centered in the Ili region of Central Asia. It was bounded on the west by the province of Shash and the Karatau Mountains with the southern area of Lake Balkhash marking the northern limit of influence.
Tughlugh Timur became the first Khan of Moghulistan and converted to Islam on the 2nd of May 1354 alongside 160,000 followers. The Moghols formed from numerous Turkic tribes including Qangli and Qipchaq as well as Mongolic tribes such as Arlat and Barin.
Yarkent Khanate finally overthrew in 1705 bringing an end to Chagatayid rule in Central Asia. Maqsud Shah died in 1930 as the last descendant of the Chagatayid house ruling the Kumul Khanate which remained a vassal of China until that year.