After my daughter was born, she just sorta laid across my tummy for a while. I remember the midwife saying, "push with the next contraction to deliver the placenta" but I never felt any more contractions. I didn't want the cord cut until after the placenta was out, which is as far as I could go with a lotus birth. I was in a really outer-space zone so it's hard to remember those first moments! OK, the midwife said, "the placenta is ready to come out, give a little push". I gave the tiniest push and the whole thing sorta plopped out. And then my husband cut the cord. I think he was holding the baby when the placenta came out. I looked at the placenta with the midwives and they froze it. It was months later that I got around to getting rid of it. I took it to the beach and threw it in the ocean.
And about eating the placenta, that's not weird at all. All other mammals, even herbivores, eat the placenta. After my sister's babies were born in the hospital, the first meal they served her was a steak dinner, and I think that's a similar thing in a way. My midwives encapsulated placenta sometimes. Hospitals sell them, some to research, but the vast majority to cosmetics companies.