ElGamal scheme signature: if private key a mod p is equal to private sig. key k mod p-1, can an attacker notice and determine the value of a? If someone is signing a document using the ElGamal signature scheme, and if the random involved in the signature $k \mod (p-1)$ is equal to $a$ (the private key), can an attacker notice? If so can he then determine the value of $a$?

Would it be because the $\beta = \alpha^a \mod p$ from the published key ($p$, $\alpha$, $\beta$) and $r = \alpha^k \mod p$ from the signed message triple ($m$,$r$,$s$) are the same?
No, it doesn't provide extra information on the private key.

Note that there is no restriction on the private key value, you just have to ensure that $gcd(k, p-1) = 1$ and that $k$ is only used ONCE. Otherwise the private key can be easily found: see this link which explains how it fails for DSA (this is the same reasoning for the ElGamal scheme).