Whatever else you do, DO NOT PUT HIM TO SLEEP IN HIS CAR SEAT ANY MORE! A recent study has shown that babies who sleep in their car seats (other than when they are in the car) are more likely to die. The angle the seat holds them in allows their heads to tip forward too much in sleep and can cut off their oxygen. Sure, it doesn't happen to most babies, but it DOES happen.
Now, onto your question... "Diff'rent strokes for diff'rent folks." I nursed my children to sleep all the time (sometimes having them sleep in the bed with me, sometimes in their crib or bassinet), and it never bothered me, they liked it, and they transitioned to their cribs just fine. I put them down for naps in their cribs, though, all the time, unless I was napping with them. I do not agree with advice to nurse your baby to sleep then wake him up for him to go back to sleep. If they happen to wake up when you transfer them, or if you transfer them when they're half-asleep, sure, pat them on the back to settle them down. In other words, you don't "have to" nurse him to sleep, but there's nothing wrong with it. In fact, my younger son would frequently stop nursing but not be tired, and push me away, and make it quite evident that he wanted to go to his own crib so he could sleep, and not sleep with me (this was in the 6-9 month range and above). I don't know who came up with the idea that moms are "supposed to" put their children down while they're awake. That may work sometimes, but a lot of times it just ends in angry and lonely children who wonder why mommy isn't cuddling them any more. Don't worry about what the "experts" say -- you're the expert on you and your son, so follow your own instincts and "mommy intuition."
My suggestion about the cough -- since being elevated helps him sleep, the "normal" breastfeeding co-sleeping position is for you to be curled around the baby with his head resting on your arm. Which means his head is elevated, and seems to be the ideal position for him to get better sleep. I'm not sure why you're wanting your son out of bed at 4 m/o, and it sounds like you get better sleep when you sleep together, so I would suggest you continue to do that -- certainly until his cough improves. If you put him down for naps in his crib, he can get accustomed to the way that feels (his crib mattress probably feels different from your mattress, at the very least), which can pave the way for the transition to sleeping in the crib at night. However, I will point out that baby humans are similar to other baby mammals, in that they need to sleep with their mothers for some time -- it helps them to regulate breathing and warmth, and allows them to night-nurse (which they still need at this age) among other things.