So he's been sleeping in the car seat throughout the night?
For the past 6 weeks he's used to sleeping in very tight and confined spaces, so the vastness of a bassinet may be scary to him. He may have gotten used to the bassinet in the very beginning.
But, really, he's just going to have to get used to it, and the pack n play bassinet attachment may not feel sturdy to him. By his age, I would just try the crib now, since those attachments aren't reccomended for long term use, so once he gets used to it, it'll be time to switch him to the crib.
Yes, he's not old enough for the cry it out method, not until he's around 6-8 months old. Crying it out at this age, will only result in a more dependant and less trusting child.
Try letting him cry for about 5-10 mintues, then go in and comfort him without picking him up (rub his back, or cheek, sing...). Then leave the room. Go back in 10-15 mintutes and continue until he is asleep, widening the time you go in each interval.
Really, he will also have to learn to self sooth. He will do fine once he gets used to his crib.
My son never sucked his thumb and never took a pacifier either, we just swaddled him tight. He slept in the bassinet until about 4 months old, but we also co-slept with him in our bed as well until he was about 8 months (b/c I would get him from the crib to nurse and we would fall asleep in bed). He transitioned very well to the crib.
I also would recommend using the swaddle wrap the other mama suggested, I hear they work great; and also use the sleep positioner. Just be sure he isn't getting too hot swaddled since the summer heat is starting to kick in. I keep a little fan blowing near my son's crib (not directly on him, but enough to keep him comfortable).