There is only one baby care center perpark, and they can be quite a walk away from wherever you are, and sometimes hard to find, even when you are close to it (Epcot and Animal Kingdom).
Quick Service and Counter Service are used interchangeably for any restaurant that you go to a counter or window or cart and order your food, and carry it away yourself. Table Service restaurants you will usually need reservations for. You check in at the podium, wait to be seated, and have a server come and take your order and bring you your food. You tip them 18% and it can be slow and crowded. Likewise tables at the counter service or quick service restaraunts can be hard to find at peak dining times.
If you are going anytime soon, you should check out allears.com for menus and an idea of prices. Also check easywdw.com for most up-to-date pictures of counter service menus, seating, and helpful tips.
Finally, Writers Stop in Hollywood Studios has a Milk Steamer, which is just heated milk. I asked them when & why people buy it. They said for baby bottles. That is the only place I have seen it on the menu. Whole milk is available many places, but getting it warmed may be a 20 minute hike to the baby center or longer.