I'm a divorced mom of 2, so I sympathize, and though I was born in the US and don't have much insight into what's best coming from overseas, I can answer some of your questions.
1. Climate: from what I've quickly looked at online, Kuwait seems to be a pretty warm climate. Minnesota is about as opposite a climate as you can imagine. It is cold there--really damn cold if you're used to a climate like what I assume Kuwait is like. Don't let anyone convince you it's not cold there. I live in Pittsburgh, PA and the winters are awful here, but Minnesota is even worse. Yet I'll hear people who grew up in cold climates constantly claiming it's "temperate" or "not so bad." They are full of it. It's bad. It's snowing right now here in November and this is not our first snow this year (that came in October). Now the summers can get hot here and in Minnesota, but the warm season is short-lived. I would say you need some sort of jacket for at least half the year, and for at least a quarter of the year, you'll be wearing mittens and hats and scarfs and shoveling snow. If you're looking for a warm climate, Southern California, Arizona, Texas or Florida are probably where you want to be.
2. Cost of living. Now, balancing cost of living with climate in the US can be tricky. The thing Minnesota and Pittsburgh have going for them is they are really cheap to live, which can be a big consideration for a single mom. California is expensive. I'm not sure about Arizona or Texas, but I'm pretty certain they are more expensive than Minnesota or Pittsburgh but probably not as expensive as California. That kind of narrows the list down to Florida. Florida has a warm but humid climate, but cost of living is pretty affordable. If I were you I would check into the University of Florida. The cost of living there in Gainesville, FL is comparable to Minnesota, the climate is pretty warm but not as oppressively hot as southern FL. I lived in FL for a year before I had kids, and more recently I've visited Gainesville, FL and it seems like it would be a good place to raise kids. Also state schools in FL are extremely affordable. To give a comparison, California schools are nearly free once you establish residency (which normally takes a year), but housing costs and food and utility costs are really high, so it's still probably cheaper overall to get your bachelor's in FL.
Just my opinion...