Shared secret beginning with 0h?
1 - Is it mathematically possible to generate a shared secret starting with 0x0n using ECDH? (My guess is "yes")


Yes; the shared secret is the $x$ coordinate; approximately half of the values between 0 and $p-1$ (where $p$ is the prime that the elliptic curve is defined over) are possible; there's no reason to believe that the values with a top nybble of 0 are all impossible.


  2 - Is there any rule or convention stating that a shared secret should be shifted so that the first nibble shall never be 0?


I've never heard of one.