1. The Bodhidharma was real person. His major role was in the introduction of dhyana(zan. chan, zen) which pointed to disciplined observation and experience of reality-rather than ritual, sutras and dependence on literature. he was not at the Shaolin temple but in a place near it.
2. Jeffrey Broughton's The Bodhidharma anthology and Red Pine's translation of The Outline of Practice sermon(labelled Two Entrances in Broughton's work)give clear indications of Chan and its roots in the Mahayana tradition also followed in Nagarjuna's work.
3. Most cultures including Chinese have had some forms of martial training.Bodhidharma's likely contribution is not into a specific shaolin style . ..but the integration of posture, breath and meditation is more likely to have informed some martial arts in China and Japan... later in parts of SE Asia.
4. memories can fade specially with the rise of nationalist ideologies but there were quite a few Indian monks
who went to China. Some for teaching, translating sutras and also for sharing and studying eclipses and ancient astronomy- see Needham's classic opus -Science and Civilization in China.Later there were Chinese and Korean monks that travelled to India and some took sutras back with them. Some Indian sutras were lost but rediscovered in Chinese translations.
5. Kalraipayyatu though well preserved is not the only martial art.In ancient Indian Buddhist centers of learning (Nalanda etc)martial training including archery was part of the curriculum as preparations for meditations.Kalaripayyatu and other Indian arts incorporated aspects of yoga into their practice. Indian martial traditions and the Chinese both have used animal names for different postures-tiger etc.at one time or another. English colonialism "contributed"(ugh) to the decline of the martial arts in India. China- while alsoa victim of colonialism-had to deal more with the economic aspects of western colonialism- whereas in India there was the take over of army and police functions by the Brits...including regulation of martial arts activities that might lead to anti state behavior.The attcks on the "thugees" are examples.
6 Buddhism declined with Hindu revivals and later with Moslem conquests- but did'nt disappear--
The Chittagong area in Bangla desh still has some Buddhists with a continuing practice..Sri Lanka as part of South Asia remained a major source of Buddhism, Tibetan buddhism was stimulated by monks from Bengal and the Ladkh area of kashmir had/has Buddhists.BTW because of Ambedkar and aspects of the Dakit movement there is ome revival of Buddhism.
And, places like the Mhabodhi society in Kolkata have always worked on Buddhist studies.
joy chaudhuri (PS. Please excuse typos)