Hi Ilovesashimi,
1. You can find the details on SQ website:
http://www.singaporeair.com/saa/en_UK/content/exp/dining/specialmeals/index.jsp
I've traveled for quite a few times with my dauther, and on SQ flights, "Baby meal" refers to those jar food, they will give those Heinz jar food. If they're busy, sometimes you need to ask from the steward/ess. Thus far, I receive only 1 jar. Sometimes they ask if I want some more. But it is not like what is stated on the web, that you'll get one each for main course, vegetable and dessert.
Since my daughter turns 1, I always book "Postweaning Meal". And the stewardess will usually give you the sticker pasted on the top of your seat and ask you to confirm you are ordering the Postweaning meal. The meal comes with 1 main course (could be potatoes, carrots, spinach or beef with potatoes, beef with spaghetti etc), 1 bottle of fresh milk (Meiji), 1 dessert (pudding), sometimes a little bit more like crackers. The main course is quite nice - depending on the menu of the day though
2. Depending on the airport and destination. In Singapore, so far we can bring water and milk for baby to the plane, not too much though and as long as you arrange them together in a baby bag. In some airports / specific destination, you are allowed to bring only say 1 bottle of water, but not more than that. Some will allow you to bring but after going thru a security check - either they ask you to open the bottle and they smell it, or they test the bottle with the water/milk on a special machine (Japan), or they take a sample of the water into a strip and bring the strip to be tested on a machine (some US airports). Others will not allow you to bring any at all.
I would say, if you don't really need to, don't bring any water in. Save you the trouble.
Since you are traveling with SQ, don't worry. You can bring an empty water bottle, then you can ask the stewardess to fill it up with water - anytime during the flight - even when you just seated. You can also bring your thermos bottle and ask them to fill up with hot water for milk.
3. For medication, I agree to bring general med fever, cough and cold, stomach pains/diarrhea, insect bites.