Only 2 weeks?! Give the whole situation time. Everyone needs to adjust to changes. I'm maternal GM to 3, paternal GM to 4 plus a GGC. I've not been at a birth or cut a cord, that's fine, both mothers wanted only their partner there. It must be an incredible experience and, as you say, a privilege, one extended to you by DD.
Practical considerations - MiL lives nearer, has more time than you, and drives. Of course she's there a lot in these early days. It almost sounds as if you object to her 'whisking away washing', did you want to do it? Her support over food and washing must take a lot of pressure off both your DD and SiL.
She took the baby out in the pram. You seem to see this as a slight to you. While she undoubtedly was proud and happy to do so maybe the couple needed a brief break from the huge responsibility of new parenthood. Recognise that the baby is fortunate to have two loving and caring GMs. It's not a competition but an opportunity for cooperation.
I don't understand your other DD's problem. She's seen the new member of your family and (if she makes up with her sister) surely will soon do so again. No one's keeping tally of who spends time with the baby on a tick chart are they?
Don't let the happy occasion of having a new baby in the family, your first grandchild, deteriorate into a bitter chalking up of scores. Forget these petty imagined slights and apparent preferences. You don't want to be harbouring resentment for years that you didn't get the first pram outing, surely? Give it time and it will all sort out.
And congratulations on your GC, look forward to the wonderful times you'll spend together over many years!