Yes, even grandparents. I was as close as it’s possible to be to my mum, but she didn’t just rock up without asking the minute I got out of hospital, and nor would I dream of doing the same to my own children, with whom I have an equally brilliant and close relationship. You have zero idea what a new first-time mum has been through, so you should take all your cues from her as to what she actually wants and needs, not assume you know best.
As a grandparent of course you don’t need to wait for a royal summons, but you should at least call ahead to check if and when it’s ok to visit. What the OP needs after a horribly traumatic birth experience is peace and quiet to recover, mental space to process what just happened to her, as well as the arrival of her baby, and time for her and DH to just bond with and enjoy their child.
Instead she’s being steamrollered by an overbearing, emotionally manipulative woman who wants to get her oar in every which way, and sulks and throws insults when things don’t go as she wants. Naturally the MIL is excited about her grandchild, which is lovely, but her approach is all wrong - everything she’s doing is about centring her own selfish wants and pushing her way in regardless. It’s just really shitty behaviour.
Having all your choices overridden, and not feeling in control of your own time and peace when you’re vulnerable and in recovery, can trigger or exacerbate symptoms of PND. But when you’re in that state it’s incredibly difficult to advocate for yourself, as you’re basically in shock and so overwhelmed that you can feel completely powerless, especially in the face of someone who’s prepared to wheedle and bully.
OP, it sounds like you’re spiralling a bit, and understandably so. Please try not to get stuck on the idea that all is lost and ruined. Your baby is only 5 weeks old. It’s such a very tiny amount of time, and you can still create your little baby bubble, but you - or more to the point, your DH - need to regain control of the situation. He needs to start advocating for you, and for the three of you as a family unit. Now is the time to re-establish boundaries you’re comfortable with and make sure he has your back every single time.
If the baby name is one you loved and would have chosen, don’t let her ‘deciding’ on it taint it for you. But if it’s ruined for you both, change the name. It’s fine. (I was a different name for about 8 weeks after I was born, but my mum was upset with her original choice and changed it!)
And sell the bloody pram. Use the one your mum bought your sister.
Congratulations on your baby 💐. I wish you peace and happiness, which you’ll undoubtedly get if your DH steps up and does what he should have done for you from the beginning.