Hi and welcome to BW.
Have you tried the wubbanub or a different paci? My son didn't take to a paci, though I wish he did, but I heard the wubbanub is easier to prop and stay in baby's mouth. But yes, as young as he is, I wouldn't worry about it being a prop just yet.
Definitely feel free to put him down earlier for nap and also feed him a bit earlier in the night...especially if you feel he is getting OT. At a week old, on-demand feeding isn't out of the question. You can certainly wait a few mins but if he protests hard, please feed him, so he can go back to sleep and so can you.