Babies should always sleep on a firm mattress because of the risk of suffocation and SIDS.
Babies that are 1 year old or under may not be able to roll themselves back over to move their face out of the soft mattress which means they could stop breathing and suffocate.
If the mattress is soft the babies face can get pushed into the mattress and then the baby can suffocate.
So always make sure the baby has a firm mattress to sleep on through the night.
After 1 year old the baby can have a soft mattress or have pillows, blankets and stuffed animals.
Yes a baby mattress for a baby bed or crib should always be firm and not soft.
The baby bed mattress should also fit the crib perfectly without gaps.
If the baby bed mattress is not firm then the babies face could get pushed into the mattress which can mean that mattress could conform to the babies face which could suffocate the baby if they were to roll over in the bed at night.
Babies 1 year and younger can safely sleep on pillows as long as you are awake and are watching the baby.
When you go to sleep then the baby should not be sleeping on pillows if they are 1 year old or younger as they can suffocate on the pillows.
Ideally though I would never recommend any babies 1 year old or under to sleep on pillows at all because the parents can easily get distracted.
Then the baby could roll over onto the soft pillows and not be able to roll back over and then they could stop breathing.
It's best to use a bassinet or put the baby to sleep or nap in a baby bed with a firm mattress and no pillows or blankets.
Once the baby gets to 1 year old or above they can then have pillows, stuffed animals and blankets in the crib.
To keep baby warm at night when it's cold outside you can dress the baby in blanket sleepers which is basically made of blanket like material that keeps the baby warm.