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.

Yes, just do not write (mod p), as it can be misleading to the reader. A mathematician doesn't care, but in RFCs they call the (mod p) groups a "prime group" to differentiate it from the Elliptic Curve group. (In fact, I think they call them "prime fields", not merely groups).

I think the keys can be smaller because every random coordinate is a valid value (valid key), but in the case of RSA, valid values are more sparse.