No it's not; it's a artifact of humans numbering items based on "total I have, including this one" rather than the more sensible but contrary to a implementation detail of human neuropsychology "number of items preceeding this one".
> "the n-th element" being at position n is handy and natural.
Yes, starting with the 0th element at position 0.
On an infinite tape, there is no "first byte", only "a byte placed where the tape already was when you started" (i.e. at offset=0.)
Similarly, there is no "first byte" of a random-access memory, because the ordering of memory is not guaranteed (i.e. some memories have bit-planes, some memories have banks, some memories are big/little-endian, etc.) The only thing you can say about a memory (and thereby about a RAM-word machine, or about an array) is what exists at address 0 of it, at address 1 of it, etc. It would be incorrect to describe address 0 as "the first byte" of memory, as this would impose a canonical iteration order.
Maybe, but human brain likes things starting at 1.
Because that's a natural mathematical model. "the n-th element" being at position n is handy and natural. What's nuts is that array indexing based on pointer offsets has made its way into nearly all high-level languages.