A group is a group. Doesn't matter whether you call it additive or multiplicative. So DH doesn't change since either way g^a and a*g are both defined as repeated action of g on g via the group operation.
ECDH has smaller keys because the attacks are (until now) weaker and not because you're using an additive cyclic group.
ECDH has smaller keys because the attacks are (until now) weaker and not because you're using an additive cyclic group.