This is too hard to answer in one post. Why? Because there were diffrent types of those interactions. While in case of Champa we had peaceful conversion from Shaivism to Islam, there was also Muhammad bin Qasim and Mahmud of Ghazni, who weren't peaceful at all.

Influences of South Asian religions are quite visible in sufi islam of India. We can read in wikipedia:

Islam was not the only religion in India contributing the mystical aspects of Sufism. The Bhakti movement also gained respect due mysticism popularity spreading through India. The Bhakti movement was a regional revival of Hinduism linking language, geography, and cultural identities through devotional deity worship.[57] This concept of "Bhakti" appeared in the Bhagavad Gita and the first sects emerged from south India been the 7th and 10th century.[57] The practices and theological standpoints were very similar to Sufism, often blurring the distinction between Hindus and Muslims. Bhakti devotees linked puja (Hinduism) to songs about saints and theories of life; they would meet often to sing and worship. The Brahman Bhaktis developed mystical philosophies similar to those advocated by Sufi saints. For example, the Bhaktis believed that there is a special reality beneath the illusion of life; this reality needs to be recognized to escape the cycle of reincarnation. Moreover, moksha, liberation from Earth is the ultimate goal in Hinduism.[58] These teachings run nearly parallel to Sufi concepts of dunya, tariqa, and akhirah.

The only book I've read on this topic is in Polish, so I suppose it wouldn't be helpful at all.

There's a little bit I can say about the expansion of Islam into Indonesia, based on reading a bunch of books on the history of Indonesia, including the Cambridge History of Southeast Asia. Islam came to Java with Arab traders. These did not set out to convert but for reasons of political and economic expedience local rulers did adopt their religion (much as local rulers had adopted the religion of Hindu and Buddhist traders before). Political and economic expedience means if a ruler adopted the religion of foreign traders, he was more likely to attract more of their trade to his lands and therefore become wealthier.

When these rulers then waged war on their non-muslim neighbors, they, being much wealthier and therefore much more powerful, tended to win, eventually driving them out, ultimately to the island of Bali, which is, to this day, still nominally Hindu.

The islamization of Indonesia (well, of Java, anyway) was therefore not a matter of forced conversion but more of a, if you will, trickle down affair. Possibly this is what accounts to the relaxed attitude towards Islam that was prevalent in Java and most of the rest of Indonesia up until the 1980s.