The Baby Killer will show up quicker if you continue to drink, do drugs, have sex etc. After your first run through, you can use the bell to the left of Cabin A to summon it. Generally though, the killer can turn up any time after the first night.
The Baby has 21HP and with each 7HP of damage you cause it, the killer will become more bloody and disfigured. Each successful hit with the Axe or Machete does 1HP. When you attack the baby with a melee attack, the Father will then appear shortly on the map. Dropping the crate will cause 7HP of damage, as will the wood chipper and any other explosion. If the weather vane falls on the killer, this will cause 8HP of damage.
The baby can be hidden from the Father once you have killed it. The trick is to not mutilate his body. Instead pick it up and throw it from the balcony in Cabin A or simply just throw it into the lake.
Don't ever use the Shotgun as it does the same amount of damage as an attack with the Axe. The Shotgun does have it's benefits though as you can use it to knock the killer down, giving you a chance to escape.
The Baby Killer's Mask. A melee attack or an explosion will always break the mask. The only way I've discovered is to setup the Rake by the woodchipper and have the Guitar handy so you can stun the Baby before running off with the mask. The Baby and the Mother will act as normal while wearing it, but the Father will not attack. This wont work if the Father discovers the remains of his offspring.