the second ability on Alex ver. 1 says that he could block illegally so does it half blacks or full block?

Quoting 2 sections of the AGR

8.3.2.6.2 - If the zone of the block doesn't match the zone of the attack, that attack is partially blocked.

and

8.3.3.2 If a partially blocked effect has been generated, the attack will deal half damage.

So it will deal half damage.