How much of an emphasis is placed on 'Spreading the Word' within the ethical framework of Hinduism or Buddism? Are there 'rewards' for bringing others into the faith? Do concepts like 'infidel' and 'heathen' exisit within the Eastern philosophies?

Nope as far as my knowledge goes both of them do not adopt the missionary approach.
While i am not an authority on all the ancient texts, the fact that an atheist movement like Charvaka was allowed to flourish around 400-600 BC (if i am not wrong similar movements were prevalent in other eastern cultures long ago) would indicate the open-mindedness adopted by these religions/cultures

No there are no such things as spreading the word in hinduism. Even if you persuade someone to become a Hindu and start worshipping the Hindu gods, you do not get any reward.

The nearest word Hinduism has to heathen and infidel is 'nastik' meaning someone who does not recognize the authority of the Vedas. There is however no such word as atheist.
But there is a word 'mleccha', meaning someone who does not follow the Hindu way of living. The Greeks and Huns for example were mlecchas when they first came to India. Then they became Hinduanized, declared themselves as belonging to some caste, started worshipping Hindu gods and therefore is considered to be Hindus.

Buddhists do spread the word, but are usually pretty low-key about it. Buddhism has a long missionary history, and the missionary stories of the great Buddhist monks are fascinating. Any good textbook on East Asian history will cover the rise of Buddhism in China.

The Hindu faith did spread throughout SE Asia by peaceful means. Hindu merchants settled in those areas and brought their priests, who often made converts among the locals. There was even a Hindu empire that spread over the area for a few centuries, but I can't remember whether it was ruled from Kanchipuram or Hampi (one of the planet's most amazing sets of ruins).

Michael