I had similar things from my mum and MIL when I first had DD (mostly my mum), and also had a similar experience on mumsnet.
My MIL also (and sometimes still does) link everything about DD to her family, which did irritate me. One day DPs aunt did it (saying DD looked just like her son as a baby, she really looked nothing like him and even MIL was amused by it) and I realised it's just a way of them looking for a connection with DD. You're the mum, and the main carer, so you've got an instant connection and bond. Other people have to try to build a relationship and sometimes get a little competitive with trying to 'see themselves' in DC (not just grandparents, my sister in law and DPs aunts all do the same!). I just learnt to brush this off with 'oh really' and it happens much less often now.
I would be very annoyed by the comments about overnight stays even if you don't want him to, you need to be firm about this when she says it 'no, MIL, DH and I have agreed that there will be no overnights for a long time yet, DS is still so little that WE aren't comfortable with it. We will consider it in a few years though' and get your DH on board to say the exact same! You need a firm, united front on this. No child that young needs to be babysat, when they're old enough to ask for a sleepover at nannys then that's different, as they actually want to! DDs nearly 2, she's never been looked after by family and it hasn't affected her confidence on her bond with them at all. She knows who her nan and grandad and aunts and uncles are and is happy to see them all. So it's not something you have to do.
Are you going back to work? I told my parents and in laws that as I knew I was going back when DD was 1 I didn't want to miss out on time with her by having her away overnight, and now I'm back at work I just say that 3 days apart from her when I'm at work is more than enough, so I still don't need her babysat. We do have issues around DDs safety with them though, (for example them not understanding that we don't want the dogs allowed to jump all over her and my dad not noticing when he's left doors open and 16 month old DD is halfway upstairs by herself). So it's more a case of not trusting them, at least whilst DD is so small.
She told me later that she thought she might have an actual panic attack when she didn't see him
Honestly, this is the part of your post which stood out to me most. It's normal to love and miss her grandchild, it's not normal to feel this way. Part of me wonders if this is to sort of guilt trip you, to make you feel like you need to give her more time with DS. Does she not have much going on in her life? You mention she doesn't work? Does she have friends? What about getting her involved in volunteer work, with children. Lots of primary schools are looking for volunteer readers to go in and read with the children every week. You could suggest that she's so good with children that it may be something she could do.
Sorry this has gotten so long, but lastly I think that when she makes comments about DS being better for her than you, it may be worth you saying outright to her that those sort of comments hurt you. You say you had a good relationship with her before, hopefully she's just got carried away with being a grandparent and hasn't stopped to consider how her these comments may be affecting you.