The local companies here all speak English (thankfully) so I didn't have a problem explaining what I wanted. We're trying to stick with American/Australian goods though and send emails to find out about cross contamination. Be careful to check that the American brands are actually manufactured in the US though, since some may be manufactured overseas.
We made several stops on the way here since it was our first long flight with our son and we didn't think he would make it (due to his allergy and being cooped up that long). We flew United and Singapore Air. United is already peanut free in coach, although biz/first class has nuts. We tried to request a peanut free flight on Singapore Air (which I heard you can do), but it didn't happen and people were being served peanuts all around us which was pretty scary. Luckily, our son isn't that sensitive to airborne peanut dust so we were ok. We packed all the food for our son for all the flights. Zojirushi and Thermos have food jars that keep your food warm for 6-8 hours which we use all the time.
There's a grocery store right next to where we're staying so we went the day after we arrived to get food. We mostly feed him fresh produce and meat which we feel comfortable with being safe. Since we're staying in a
Service Apartment so we can cook all his meals. We did pack a bunch of snacks with us until we could find safe ones here and also air shipped a lot of dry goods that we weren't sure if we would be able to find here or not. We were a little worried they would be stopped in customs, but we got everything ok.
Hope that answers all your questions! Feel free to email if anything else comes up. Good luck with your trip!