I remember as a JW asking Jehovah many time to find my keys so I wouldn't be late for field service or work. Sometimes I had to ask 2 or 3 time but it never failed. Jehovah always made sure I found those keys.
I've asked Jehovah to help me thread the eye of a very small needle (in jesus name of course) and sure enough with Jehovah's help I did it.
My wife used to ask for some good Bible studies like the ones you read about in the WT and Awake but he wasn't to good at granting her request which made her depressed and a little mad at Jehovah. Maybe that's why he didn't answer her prayer,, because Jehovah don't like it when you get mad at him and so he witholds his blessing,, I mean if he will wipe out the whole world of mankind because they are not worshipping him, or are not worshipping him the "exact" right way then I suppose if you even get a little mad at him,, there goes any possiblity of him answering your prayer even if they are in Jesus name.
I don't think Jehovah will answer any prayer for world peace because of the issues raised in the garden of Eden. Jehovah wants to show man that they must worship him "exactly" the way he says and that if they ignore him in anyway he will curse them and make a mess of everything they do. And so that is why prayers for finding keys and other little helpful things Jehovah will grant,, because these teach man to always look to him even the very little things in life. Jehovah doesn't want you to be independant so the real bad things like war and starvation Jehovah has not put an end to,, in order to teach us all to not put our trust in man in whom no salvation belongs,, and to keep praying to Him to find your keys.