Short answers, pulled from the story.
The first Muslim naval expedition to India occurred in 636 AD under Uthman ibn Abi al-As al-Thaqafi, governor of Bahrain and Oman. This raid targeted Thane near modern Mumbai without Caliph Umar's permission.
Muhammad bin Qasim departed Shiraz with an army of 6,000 Syrian cavalry and 3,000 camels by 710 AD to conquer Sindh. He defeated Raja Dahir at the Battle of Aror in 712 AD, establishing Arab rule as far as the western shore of the Indus River.
Mahmud of Ghazni ascended the throne in 998 AD and launched seventeen expeditions into northern India. Lahore became his second capital and later his only capital after 1030.
Timur began his campaign against Sultan Nasir-u Din Mehmud of the Tughlaq Dynasty in late 1398 and defeated the Sultan's army on the 17th of December 1398 at the gates of Delhi. The city was sacked and left in ruins before Timur departed in January 1399.
Babur founded the Mughal Empire in 1526 after defeating Ibrahim Lodi at Panipat. He died shortly thereafter in 1530 but established a dynasty lasting over two centuries.