For the first part of your question the answer from the academic community is pretty much a universal no, there is no (reliable) evidence. The people who do support the "Jesus in India" hypothesis tend to be outside mainstream academia and perceived as something like amateur conspiracy theorists.

I thought I'd look into this a bit more to check as I am familiar with the idea through some of the later proponents. It's the kind of topic where there's more information about it on Wikipedia (Nicholas Notovich) than there is in my university library. A search on the database through up a whole 12 results, only two of which were relevant, and one of those was Notovich's book. This leaves one article that addresses the subject, "The post-truth Gospel" by Marcel Theroux (2018). It's not so much an academic article as an essay for the Times Literary Supplement, which is a UK newspaper. Some more serious biblical scholars, like Bart Ehrman have commented on it, negatively, so overall it's not made a huge impact on the world of academia.

Curiously enough though, one of the points made in the Wikipedia article refers to the phenomenon that Notovich is dismissed in Europe but retains some popularity in the US and India. As it happens, I lived in India for a short time and while I was there I picked up copies of Holger Kersten's Jesus Lived in India (1983) and Elizabeth Clare Prophet's The Lost Years of Jesus (1987). I got these and a couple of others on related themes from a mainstream bookshop the which does support the claim of greater popularity somewhat.

Reading Jesus Lived in India is a pretty wild ride. Amongst the claims that Kersten makes are that the Jewish exiles from Egypt went to the Punjab region during their wilderness years and that Jesus learnt a special technique to appear dead on the cross so faked his death in a sense. There's also extensive discussion on the shroud of Turin.

The Lost Years of Jesus is not that different. It's presented as a murder mystery for the reader to solve and the accounts of Notovich and a few other travellers are given to help the reader make up their mind. The author does present many of the criticisms of Notovich, which is good, but sections of it read like The Secret - that law of attraction, self-help book - in the way that numerous quotes of believers are found throughout, most of them mysterious in their provenance.

Some of the main criticisms of these accounts are that the writers pushing their narratives have very shaky methodologies, if any, and are often too credulous of things that indirectly impact on their theories but that suggest a general lack of critical awareness. The Turin shroud for example, or even accepting that Jesus had lost years (the implication is that the birth narratives are largely correct - the subtitle to the book is 'documentary evidence of Jesus' 17-year journey to India, Nepal, and Tibet').

Most scholars of the historical Jesus claim that either there's very little we can say with confidence about Jesus when it comes to even modest accounts that fit into a first century Judaean context, so throwing their weight behind a much more fanciful story like this is never going to wash with them. Those who believe the gospel accounts are largely accurate (usually evangelical Christians) are happy to accept the carpenter hypothesis and would also find the India notion too far fetched.

However, strictly speaking, and more directly related to your second question, certain aspects are technically possible, albeit highly unlikely. There was some contact between India and the Roman Empire and there were both Jewish communities in India and Buddhists who occasionally show up in the Mediterranean area. On that basis alone a Jewish individual could have met a Buddhist or two, and this would be even more likely if they were a trader, but it doesn't seem very likely that Jesus, a Jewish man from the Judean countryside, would have naturally come into contact with Buddhist travellers.