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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.

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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.

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