The air in Delhi is filthy. Many of the older cars, buses and trucks have diesel engines that spew out filthy black fumes. Many of the scooters and auto-rickshaws have two stroke engines, so they also burn a lot of oil. Add to this in the winter, there is a temperature inversion layer over the city that traps the pollution in the city and the street people burn rubbish and cow-dung to keep themselves warm.
I worked there for two years and came back with what was diagnosed as asthma, but wasn't and cleared up after about six months. I can't honestly say that I would recommend it for children with asthma, much as I enjoyed some aspects of the life there.
At the very least, you need to take a trip out and see for yourself. It is a mad, mad, mad city. If you've never been to that part of the world before, prepare yourself for a shock regarding the level of poverty you will see, the hassle you will get from beggars, seeing lepers (a perfectly treatable disease) missing body parts, the noise, the smells. Many people experience a level of culture shock that just makes them want to go home.
On the peanuts side, I am not sure that peanut oil is used much in cooking in Delhi, but it is used as a cooking oil in other parts of India. Generally ghee (clarified butter) is used in the north of the country. However, you would have no way of finding out if eating out. The sideways head nod would be the answer to the question - it has a variety of meanings, but generally I found it to mean "I am in total agreement that you have asked me a question". Food labeling and commercial foor production standards are pretty much non-existent - while they may exist on paper, compliance certificates are for sale, so you would always be suspicious of what you were buying for home cooking.
I would say that the worries about your children's health would far outweight the delights of Delhi - or more specifically the delights of the rest of the north of India, since, in my opinion, Delhi has nothing to recommend it and is my second least favourite of the cities I visited - Agra being the worst one.