You can look at
GreatSchools.net for school information. I love this website as it gives so much information. I totally agree with most regarding NKC - it is a great area with more diversity than many other areas. Liberty is nice, not that diverse but I think it is starting to change, the schools are also great.
Lathrop imo would be a very scary move for your children. I do have a friend who lives not to far from Lathrop in Lawson and her youngest is mixed and she says she feels totally fine with no tension or racism. She says there are many mixed kids there, not black kids, but mixed?? No clue about that just letting you know her experience. Lawson has good schools and my friend who lived in the inner city of KC for a long time said her children have all thrived since moving. She loves it! The drive however is tough especially with gas prices. I don't get a creepy backwoods feel and economically it seems very diverse - it is a cute, affordable town. Honestly I would call the school and talk to them if you really want to explore living outside of the metro area.
Many neighborhoods do have HOA fees and normally that will include pool, parks, trails and making sure people follow the rules like not painting your house purple and orange and maybe trash pickup and possibly snow removal. Most of the time they aren't that much unless it is a townhome in that case the fees can also include your homeowners insurance and lawn care.
I wish you and your family all the luck. I know how hard it is to try to do the best by your children and to constantly worry about every decision you make and wonder about the impact it can have on them. It just means you are a great mom.
All and all I have to say while we have our challenges here it is a really nice, family oriented place that is changing and evolving constantly. I think there is a strong possibility you all will love KC and all that it has to offer.
Take care
Jen