You can check in the baby pram on top of your luggage. Or if you like, you can even push your baby in the stroller all the way to the boarding gate, and they will keep it aside for you and you will get it back when you get off the plane.
A slight caveat, my neighbour's stroller was broken when she did this, but she didn't claim . Mine was fine, though.
Infants are given 10 kg checked baggage allowance on top of their stroller.
SQ generally will take quite good care of children, but it does seem to vary from flight to flight. I think for infants they have a little package with diapers and wipes, and I remember someone saying they will even give you baby formula.
I've never been to Bangladesh, so I can't tell you what is 'must-bring' from here. If you know ppl there, maybe you can ask about diapers (very bulky and difficult to bring from SG).