Translation:He makes the children happy by making them eat vegetables.
Yes and no. You can probably use it and be understood as you expect. However, if you look at it closely, you will notice that the guy could make the children happy (making tricks, telling funny stories...) at the same time (while), but in parallel, so independently from having kids eat vegetables.
In English, using the word "by" indicates causality. So unless that is the unambiguous intent of the speaker, "while" (in the sense of "during") would be more appropriate here. In either case as you have described, the happiness does not come from eating vegetables, but from a separate, unspecified action. The translation Duo provides, "He makes the children happy by having them eat vegetables" is unambiguous in English; it means that the act of eating vegetables makes them happy. That would not seem to be the case in the French.
1) Would somone please help me understand why "leur" is used instead of "les." I have forgotten why we use "leur." <<Il les fait laver la voiture.>> = He has them wash the car? It has to do with using the gerund?
2) Isn't the "their" omitted in English? By their eating some vegetables, they are made happy by him. By eating some vegetables, they are made happy by him. In eating some vegetables, they are made happy by him. By having them eat some vegetables, he makes them happy. (It really feels like "les" to me.) Having to eat some vegetables, to them, is a happy moment, one especially for him.
3) In French, is the "il" included in the "leur" or not? The "il" could be another child.
When a causative verb (faire manger - to make eat/ to have eat) has a receiver (des légumes) and an agent (les enfants), the agent is indicated with an indirect pronoun. For example:
Je le fais manger -> I make him eat
Je lui fais manger la pomme -> I make him eat the apple
Je la lui fais manger -> I make him eat it [the apple]
We cannot use "their" here in English, actually, because "faire ~" is a causative verb. "He made them happy" is how it should be, because it is the "il" taking the action, NOT the children.
I can't definitively answer your third question. The answer is most likely "probably not but it depends on context". For the most part in English you can't use a causative verb on yourself except in cases where you're expressing duality of intention (eg "I made myself eat the bitter gourd, even though I didn't want to.") It may or may not be the same in French.
You may want to read up on causative verbs. http://french.about.com/od/grammar/a/causative.htm
Maybe you should get your ears trained to differentiate nasal sounds; un, an, in, on
Try this and tell me if you hear them clearly: https://translate.google.fr/?hl=fr&tab=wT#fr/en/un%20grand%20pain%20rond
If the French meaning is that by making them happy the children then eat their vegetables it shouldn't use the by, because it does indicate in English that eating their vegetables is making them happy. Instead, the English translation should be "He makes the children happy to get them to eat their vegetables, or so they eat vegetables. I don't quite understand why literal translations are used that don't convey the correct (same) meaning in both languages.
to make someone do something = faire faire quelque chose à quelqu'un.
so when the "quelqu'un" is represented by a personal pronoun, you have to use its indirect form:
- me faire manger
- te faire manger
- lui faire manger (= à+il or à+elle))
- nous faire manger
- vous faire manger
- leur faire manger (= à+ils or à+elles)