The thing that struck me in the OP's post was that 'now he's started talking about how, when I'm on maternity leave, I can start picking up dsd from school'.
Her dp hasn't asked her if she would so they can have a proper conversation and work through they whys and wherefores of this. He has just assumed that maternity leave = not having anything else to do and that therefore he can dictate something for her to do in that time.
There's another thread running at the moment about a dad assuming that his partner - currently on maternity leave - will look after his 9yr old dd while she is on half term as neither he nor his ex can look after her - he just said she would without checking with his partner that she didn't have any plans but assumed it would be ok (it was when the ex was supposed to be looking after her). The consensus on that thread was very much that the dad was being unreasonable in expecting his dp to provide the childcare without asking - if nothing else it's just plain rude to make plans for another adult without checking with them first - step-parent or not.
Having a new baby is very different from caring for a dsd. You don't know how your own body is going to react - you might spring back and be fine after a couple of days (don't we all wish!) or you might end up with a c section that has a really painful scar, problems walking and driving, a baby that needs to stay in hospital, PND that means you just want to hide all day, a baby that screams all night and happens to sleep for a couple of hours mid afternoon so that's the only time you manage to catch up on your sleep etc at the other end of the extreme. And most of us fall into the middle somewhere.
The only thing that you can say for definite is that you shouldn't be making any definite plans for after your new baby's birth.
Have a long conversation with your dp - need to crack on the head anything that makes him think that he can dictate anything that you do with your time!
Once that's done, chat through the picking up dsd situation. Point out that if it was your own child, you would probably be leaving her at the childminder - at least until you were more settled and knew how you were coping. And that once you know how you are coping you will then be in a position to make better decisions about how and when you can help out...
I know when I had ds1 I found it very difficult to do things at a specific time because there would always be something that cropped up to thwart it - a mega poo explosion or a screaming baby that wouldn't eat but had just started and you wanted them to eat more, or we'd both be so tired we'd oversleep or whatever. I was lucky that I didn't have many things that i HAD to do at specific times - just a few doctor's appointments - and getting to them on time was hell, despite always starting out with the best of intentions. Knowing that I had to be out of the house on time to do a school pick up 3 days a week would have been a nightmare. (and yes, I know plenty of people manage to do this day in, day out. I'm just saying that for me, at that time, with lots of other horrible stuff going on at the same time, it would have been a nightmare. Other people would have been fine with it but might have had problems coping with stuff that was managing to cope with, you just don't know until you are there).
It is scary when you are looking into the future and you try to imagine how it will be but you don't actually know. But try to take some comfort from the fact there are lots of others out there that have been there before you, and lots of them are on MN and more than happy to hide from their own dc share their experiences and say how they managed to cope, which hopefully will be able to provide you with some support!