My Baby Does Not Sleep Through The Night. He Wears

My baby does not sleep through the night. He wears cotton sleeps in the same bed next to me sometimes at different distances. He won't keep down his formula and we are now on so lactose free. During pregnancy I was diagnosed with cancer. But I did not take any major medications until my third trimester. The baby was born at 4 pounds 5 ounces. Now at a month old he's weighing just over 5 pounds. I have tried letting him cry himself to sleep but as you said if that doesn't work. I've tried walking him, cradling him, Rockingham and everything else every time I feed him he spits up almost half of what he eats on the plug only works for a moment it's getting it out of his mouth is that he's so jittery that he knocks it out. I use a calm voice and talk to him like normal. He also has a trouble when he is having bowel movements and seems to go right it's almost as if it hurts. But his stool is runny. Is there something I'm doing wrong? How can I get him to sleep through the night keep his formula down and comfort him when he's having a bowel movement?