Where did Hinduism spread?
Maurya Asoka converted to Buddhism after witnessing a bloody battle and sent missionaries to spread the word.
Beginning in the first half of the 1st millennium ce, many of the early kingdoms in Southeast Asia adopted and adapted specific Hindu texts, theologies, rituals, architectural styles, and forms of social organization that suited their historical and social conditions. It is not clear whether this presence came about primarily through slow immigration and settlement by key personnel from India or through visits to India by Southeast Asians who took elements of Indian culture back home. Hindu and Buddhist traders, priests, and, occasionally, princes traveled to Southeast Asia from India in the first few centuries of the Common Era and eventually settled there. Enormous temples to Shiva and Vishnu were built in the ancient Khmer empire, attesting to the power and prestige of Hindu traditions in the region. Angkor Wat, built in the 12th century in what is now Cambodia, was originally consecrated to Vishnu, although it was soon converted to (and is still in use as) a Buddhist temple.
Hinduism is more populous, but Buddhism is found in more different places and practiced by more diverse groups of people. It is important to note here, the difference between active proselytizing and passive spreading. In the case of Hinduism, the religion has not been actively proselytized outside of India for nearly 1500 years. That Hinduism has spread outside of India (to far off places like Fiji or Guyana) is because the British Empire transported Indians…
Did the Gupta's make Hinduism the official religion and build many a Hindu shrines to their deities?
Yes. Hinduism is beleived to have had its origins in the Aryan culture of southern Russia, from where it spread to India early in the second millennium BCE. From India, it spread across Indonesia, only to be supplanted by Islam everywhere except the island of Bali. Indian labourers carried Hinduism to Fiji, under British rule. With the recent migration of Indians to all parts of the world, they have taken Hinduism with them everywhere.