I have two extremely fluffy house cats, their hair gets everywhere. I do hoover as often as I can but impossible to get rid of so I just had to kinda roll with it! I imagine my newborn inhaled a lot of hair, we all do!
I kept the plastic cover the mattress of the crib came in & popped it back over during the day.. one of my cats did like sleeping in it but absolutely not when baby was in there! They never bothered with the pram.. they actually used to sleep in the basket underneath the pram 😄 but I did just take the carry cot/seat off the frame and leave it upside down when not using it..
We kept them out of our bedroom, just shut the door. They eventually got bored & stopped meowing! It’s actually easier now we have stair-gates as they can’t jump them (useless). I’d possibly start implementing that rule now, so it’s less of a double whammy!
The other we did was buy them a new, quite tall cat tree so they could both escape to safety & even now, they sleep on it more often than not!
Oh the whole, we found both cats were very cautious & wary around DD. They hated it when she cried & would keep their distance! They never went near her Moses basket or say the playmat while she was on/in it.. She’s 2.5 now & one still won’t go anywhere near the bonkers toddler!
I think start with the notion to never leave your newborn alone with them & judge it by how they are reacting. I eventually felt comfortable to nip & make a quick sandwich while she was snoozing etc..