I'm a home daycare provider and a mom to a 3 month old, so I have a few suggestions. It really depends on which is more important to you at this age: teaching good, consisten sleep habits or where the baby sleeps.
With the daycare kids, I let them nap in a swing or a bouncy chair when they're that little. Everyone takes their afternoon nap at the same time, so it's easier to train them to nap when I want them to in a swing/chair and I'm able to keep the little babies near me so I can get to them quickly without waking the other kids. Swings/chairs are a place of comfort for them and they can sleep and wake and sleep again without feeling alone or getting too fussy. Besides, I'm just not comfortable letting the tiny ones out of my sight, even with a monitor. My daughter takes her morning nap in her bouncy during the week and her afternoon naps are always in her swing for consistency. At night, we've always put our daughter in her bassinet to sleep. She naps in there on weekend and Daddy mornings.
If your daycare provider has a swing (which they should if they care for infants), let him nap in the swing, since that's what he's comfortable with. He'll have enough to adjust to (no Mommy/Daddy, new people/sounds/smells, schedule changes) that there's no reason to upset another aspect of his life right now.
If you want to get him out of the swing at night, you might want to go back to swaddling him. He might like the swing for the incline, but he might also like it for the snug feeling. Swaddling will give him the comfort of snugness and keep him from feeling uncomfortable in the open bassinet. Also, try putting him in his bassinet before he's totally asleep. Let him fall asleep on his own in the bassinet and he'll get used to sleeping there quickly. I'm not suggesting you put him down wide awake and let him scream for hours, but hold him until he gets drowsy and put him down before he's fully asleep. Play music for him after you lay him down and start a routine that he can learn means sleep, just as he's learned that the swing means sleep. If you start now, he'll quickly learn that getting his diaper changed, jammies on, being held/rocked for a bit means sleepy time and will soon be able to sleep where ever you put him when that routine is followed. It's how I've taught the daycare kids, ranging in age from infants to preschoolers, to nap all together and at the same time everyday and it's how I've taught my daughter to nap/sleep under any circumstances.
As for your last question about what to cover him with, a company called Halo makes a SleepSack that is endorsed by First Candle/SIDS Alliance. Basically, it's a vest-like sack that is a wearable blanket. It zips top to bottom for easier diaper changes, comes in many sizes and colors and is available in a SwaddleSack version with a detachable, velcro swaddle.
Good luck!