Re: Spit up and choking
I agree w/ Haley's mom- ask your pedi about possibility of reflux if your LO is having fussiness along with the spitting up. The burping thing will get better- probably a combo of just learning to nurse and swallowing lots of air + crying. It should get better around 6-8 weeks once he has BF'ing down.
I never used a nipple shield, so I don't know if that makes burping worse. Lots of people here have so they will give you good advice if that makes extra burps for baby.
The wheezing/gurgling sound you hear could be saliva pooling in the back of his mouth, that too should stop in a few weeks. I agree to not use bottles/paci's until you have the BF'ing down which it sounds like you do.
My LO sleeps on his side or stomach, he never liked sleeping on his back so we never forced it. I also co-sleep so I always know what's going on with him during the night.
If you're tired really think of co-sleeping and if that doesn't work for you try and get as many naps as possible when your baby is napping. Take good care of yourself and sounds like you are on the road to a beautiful BF'ing relationship!!