The car seat is definitely not OK for all night. I think you might just have to work at it to get him to settle. He will get used to it after a wee bit of time, but I would suspect there isn't an easy way to do it.
When you say you've raised his head, do you mean raising the matress or the cot? If there is a problem with reflux, raising the cot 5 or 6 inches at one end will help.
If you lay him down and he cries, try to settle him for a few minutes, soothing tones, loud shusshing and patting often work. But if it doesn't work, pick him up to settle him and try again. At first this will take many, many attempts, but it shouldn't take too long before he gets used to the fact that the cot is where he goes to sleep.
There is another option, where you leave him lying down and try to settle him using touch and voice, without lifting him. Again it will be difficult the first few times, you could be there for an hour, but it will get less and less very quickly. If you're not used to leaving your baby to cry for a few minutes and are more likely to pick him up every time he cries, the first option is likely to be easier for you to handle. Certainly for a newborn, the pick up and put down is likely to be more successful.
We used the second option with Abby and within a week, she was going down into her cot with minimum fuss at night time. Naps took a bit longer, but now she goes down with no fuss at all. We found that helping her find her hands, bringing them to her face was very useful. She now does it herself whenever she is sleepy.
