I don't have dogs, but my sister does. She has lived abroad in several different countries and brought her dogs from the US. So, from her, I've learned some things.
Be sure you get the European chip; she had to have her dogs "chipped" twice, even though she had requested the correct kind the first time. As she likes to repeat, if you're filling out the paperwork and the number of digits on the chip doesn't match the number of digits on the form, you've got the wrong chip.
Each airline has different regulations, and these keep changing. So, you have to check the airline(s) you are considering for their latest rules (and fees, which keep increasing). Similarly, check carefully on carrier sizes required (she once had to buy a new carrier just because of this; her previous allowed carrier was now a tiny bit too large).
There are limits to how many dogs can be carried on each flight; be sure to get your reservations in early so your dog is not "shut out" of your flight.
I don't know what she did about food or water before the flight. Her dogs were too large to go in the cabin, so they went as "baggage," and did fine. But, she emphasizes, under no circumstances would she let them go as "cargo" (the third option).