I don't know anywhere where the Koran goes into any detail about Christian beliefs. The "Holy Spirit" appears alone four times (in the translation used at the University of Michigan site), three times in connection with "strengthening" Isa (2: 87, 2: 253, and 5: 110), and once as the vehicle of revelation (16: 102), that one possibly a reference to Gabriel personally.
Just plain "Spirit" seems to parallel Jewish usage: God's action in time and space. You can use word or phrase search at the University of Michigan site I gave earler. Obviously, that is limited by the translation.
I am unaware that Mohammed had much Christian opposition to speak of. He is mostly busy converting pagans to his variety of Abrahamic faith. His successors surely found use for Christian counterapologetics, but Mohammed had immediate local concerns.
You keep saying that. I have also objected to long ellipsis within the same sermon.
In either case, textual separation is supportive of, but not the basis of, the inference it supports. I infer that Mohammed is talking about different things in different places, because his descriptions are different. That the "places" are not adjacent is a reasonable follow-on observation to make. In each case, I chose to describe just how non-adjacent they are. So what?
53: 19ff surely are. Allah is not part of the Arab pantheon. Elsewhere in 53, Allah emphasizes, at length, his continuity with the Jewish God. The theory of Islam is that it does not institute Arab monotheism, nor does it import it from elsewhere, but rather restores it. The truly Arabic true God isn't suddenly One, but rather has always been One, and Seventh Century Arab pagans' ancestors had gone astray.
The book would certainly be a lot shorter if all Mohammed had to say was that there is no god but God, and Mohammed is his messenger. That statement plainly calls for some elaboration and specification, not just repetition of it, but that statement is the key to the whole book, IMO.