My questions are:
1) What kitchen utensils are important to bring over? (Silverware, glasses, plates/bowls,etc).
Whatever you bring, bring less than you think you need because the "kitchens" in Hong Kong aren't typically what you'd of being an American. Lack of storage is going to be an issue. If you need the odd serving platter or tongs or what not, those are easily sourced from Ikea or the many kitchen/restaurant stores on Shanghai Street in Kowloon. Keep in mind most people don't entertain at home like we do in the 'States so 10 place settings would just take up space and never be used. Entertainment is done outside of the house.
2) What about cooking utensils? (pots, pans, pyrex dishes, baking sheets, spatulas,etc).
I would guess 1 in 10 flats have an oven. If it's important to you and your family, be sure to specify to the real estate agents you are working with you require an oven. Also keep in mind there's a pretty good chance that you'll hire a helper (maid) and the helper will end up doing much of the cooking. So if you are protective of your Le Crueset or whatever, leave it at home.
3) Are we allowed to bring medicine (aspirin, allergy medicine, etc) in? If so, how much is allowed?
Yes, I don't know technically what you are allowed, but I can't imagine immigration caring one bit if you have a couple of jugs of Costco-branded antihistamines. You'll want to carry meds. A six week journey on a ship that may or may not be climate controlled would not be good for your meds. I couldn't find Pepto in HKG so I always had some of that around.
4) What are some other necessary items to bring over that will help us feel more at home?
People with big feet have trouble finding shoes in HKG.
It's a fine balance between being comfortable in a new place vs. trying to recreate what you have at home. I'd recommend against the latter. Otherwise what's the point of living abroad?
There are many threads asking the questions you've asked. Might want to seek those out.