MissDuke - if what you are bolding has line breaks/paragraph breaks in it, you need to put the asterisk at the beginning and end of each separate line, rather than just at the beginning and end of the whole post.
I think it is a lovely idea for a grandparent or other relative to have the baby overnight, to give an exhausted new mum a break - but only if the parents are happy with this, and actively want this arrangement (so the MIL in the OP who is pressuring the OP - that is wrong, and the OP shouldn't be forced to consider this arrangement), and if the relative looking after the baby sticks like glue to the way the parents want things done (so grandma co-sleeping, when she had been asked to use the Moses basket - that is a real no-no too).
I do think it is possible for a mother to sleep through her child's crying - if she is tired enough. A friend told me about waking up to find her dh desperately trying to latch the baby onto her boob - and this had been going on for a while - she was just so tired she didn't wake up. I would worry about anyone co-sleeping with a baby, if they slept that deeply.
I have looked after a friend's baby overnight - she and her dh were totally exhausted, so I went and stayed over with them, and had the baby in with me, to give them a night's sleep. But the baby wasn't a newborn and wasn't breastfeeding any more. And I didn't pressure the parents into this arrangement - I offered to help, but if they had said no thanks, I would have accepted that without a murmur, and would have asked if I could do something else to help instead.