The last time I travelled it was myself with my almost 2 year old (in a seat) and my 7 week old (on my lap), on a 12-hour international flight.
1. You CAN bring your own juice/formula/milk for a child under age 2. Just make sure to bring it in a bottle, and NO WATER! Water you can get anyway at the airport (any restaurant will fill a cup for you if you ask nicely), so that shouldn't be a big deal.
2. When I travel with my kids I bring my own carseat, but that's because they have always still been in infant seats, and I want the one that snaps onto the stroller. Since your daughter is not yet forward-facing, you might want to bring your own, unless she's big enough for the 2-in-1, which is what most rental companies have.
3. I always took my stroller, because I can use it right up to the gate, and that way I don't have to carry all the bags (diaper bag, carry on, carrier). I do take a carrier/Snuggli as well in case my kids decide they want to be held the whole trip, as it leaves my hands free for other things. I also don't have to juggle the bags and the baby while trying to get the luggage of the carousel...I can leave the baby in the stroller (next to me at the carousel) and have both hands free to grab suitcases.
I never saw a need to travel with a pack 'n play, so I can't help you with that.
Pack light! It is entirely possible to pack for 2 people for a week in one large suitcase. Keep in mind that you have to be able to lift it! I never packed diapers (I had a diaper bag for the plane, and I packed 1 diaper for every 2 hours we'd be away from the house...including travel to and from the airport, waiting times, and flight). You can buy diapers when you get where you're going. Same with wipes. One stroller & one rolling suitcase is definitely manageable.
You have 3 pieces of luggage allowed, unless your ticket says otherwise, and your stroller is NOT counted as one of them. Your carseat will be if you are checking your stroller rather than taking it to the gate. Use your allowance wisely!
I also always had 2 carryon bags: one real carryon (snacks, books, toys, music...) and one diaper bag. My diaper bag always had diapers & wipes, a change of clothes for each of us, my breastpump, and any valuables. It could have accurately been called my "oh my G-d they lost my luggage" bag.