1) Similarities could be found in Chinese, Tibetan, Sri Lankan, etc. dance, music, martial arts with Indian arts. E.g. Therukootthu, Kathakali (flamboyant facial make-ups), Kerala Chenda drums, the rhythm all are adopted in China, Tibet, Japan and other south-eastern countries in the form of Lion, Dragon dances' flamboyant decorations, drums, rhythm, etc.
2) Similarities could be found between martial arts and Mudras (symbolic gestures) of yoga & Indian classical dances, and their movements. Probably dance, martial arts and yoga all might have evolved at the same time along with music, meditation, etc. They share the common property of being melodic.
There are several categories in Varma: NOkku (vision) Varmam, Thattu Varmam (deals with treatments and defense based on vital points), Chandra (moon) Varmam, Surya (sun) Varmam, usage of weapons on vital points, Varma treatments for animals, etc. In local tongue often Varma is called as "Marma", which may not be correct. The word "Varma" doesn't have a Sanskrit origin as in "Varn(m)a - color". Varumam in Tamil means vital points (there are other meanings too).
These techniques are extensively developed by Siddhas (well known for their Ayurvedic and alchemy works), who dwelt in mountains of South India, especially Tamil Nadu. Even today, there are legends around Courtralam, Suruli, Pazhani and Nilgiri hills, which speak about surviving Siddhas till date, who have eventually achieved longevity through their knowledge in Ayurveda, alchemy, meditation, etc.
Sadly, many of these techniques are lost due to the Guru's fear of misuse on co-species (as we humans are notorious for turning any scientific works, bar none, into weapons of destruction), and thus they refused to transfer the knowledge to others, including to their own descendants. This happened particularly, when Buddhism turned into political movement and the Buddhists started to learn these arts for their political gains.
However tiny fragments of Varma survived in the form of literature, mainly Tamil & Malayalam, and got the attention of Gurus in Kerala, a small state in south India, arround ~15-17th centuries, and became Kalarippayattu. This famous traditional training of Kalari of Kerala, is always done inside the Kalari (literally, threshing floor or battlefield), which is a specially constructed practice area. Payattu means 'exercise in arms or practice'.
A few centuries back in Kerala, quarrels between local kings were resolved by fixing an Ankam (war), a duel to the death, between two Ankachekavars (kalari fighters), each ruler being represented by one Ankachekavar. The ruler represented by the surviving Ankachekavar was considered the winner.