Hi from Missouri!
I just might have the information you are looking for. DH and I currently live in Columbia, Missouri (about 30 minutes from Jefferson City), prior to here we lived for 4 years in Branson and I lived near the Lake of the Ozarks about 10 years ago.
I guess I will just take each on of your points and try to give you my opinion.
I was doing a little research this week, and it seems like Jefferson City is a nice, quiet town.
Jeff City is the capital of our state but it still not a very large town. Most of the town is focused on the state government (obviously) and right now jobs are not great there because our state is having major financial problems. The job market in Jeff is not doing the best right now simply because probably 80%+ of the jobs are state jobs. We have been to Jeff lots of time since it is just 30 minutes from where we live but they just don't have near as much to offer in the way of housing, shopping, etc. A great deal of the people that work in Jeff actually commute from Columbia since it is a very easy drive down 63.
We were looking at the Lake of the Ozarks area and Branson, too, seeing as I am majoring in Tourism/Hospitality and it seems there would be a lot of jobs available in those areas.
The Branson area is certainly booming and there are lots of jobs available in the tourism industry but the vast majority are very low paying jobs and the housing costs are quite high in that area. Most people are also out of a job during the winter months after Christmas when tourism dies down until the spring. The traffic is TERRIBLE (except in January/early Feb). You learn to live with it when you live there but you must learn to have a high tolerance for idiot drivers if you move there. On the upside the area is beautiful and if you like outdoorsy type stuff there is a great deal available. Springfield is the 3rd largest city in the state and it is about 35-40 minutes away from Branson on Highway 65. If I was looking to move to the Ozark area I would really look at Springfield or some of the smaller towns between Spgfld and Branson (Nixa/Sparta) rather than Branson proper. You still get the benefit of living in the Ozarks and close to the touristy areas but Spgfld is so much larger and has more to offer for the locals as far as housing, etc.
The Lake area is a nice area that is just 30 minutes south of Jeff. It is also a beautiful area that is very touristy but not to the extreme level of Branson. Alot of the jobs are seasonal like Branson but the traffic is not as bad. Housing in the immediate area is quite high as well but there is affordable housing in the outlying areas.
What about climate? Does central and southern MO ever get snow? We would like a little snow now and then. Not Christmas without it.
The climate varies. The joke is that if you don't like the weather now just wait 5 minutes. We are cold in the winter and hot and humid in the summer. We do get snow in the winter most years at least 2-3 times although usually not more than 4-5 inches at a time but every now and then we really get dumped on with a foot of snow or more.

This winter has been very snowy and we seem to have had snow every week here in central Missouri. SW Missouri (Springfield area) has really been hit hard this winter with several 12"+ snows and ice. The snow usually doesn't stick around all winter and while white Christmas do happen usually our biggest snows come in Jan/Feb.
I would like to move somewhere the crime is low (moving from MSP, so comparable to that or less), schools are good, job market is good.
I would have you look at Columbia (and not just because I live here). The job market is good here and we usually have the lowest unemployment rates in the state and even the country. The University of Missouri's main campus in located here in Columbia and they are the biggest employer in the area. They have jobs in just about every field. There are also two other colleges in town. Having the colleges here provides nice cultural facilities, sports facilities, etc. that you would not always get in a town of our size (125,000).
The crime rate is low and we have never felt like we were in any kind of danger. There are a few areas of town that I would not want to be in after dark but they are isolated and I think that is a problem everywhere you may live. The schools here in Columbia are good. Most of my friends who have kids in the schools here seem happy.
Housing is quite reasonable IMHO. We just bought a nice 1 year old 3bd/2bth house for $111,000 and there is lots of housing available both below and above that amount. We live in a great subdivision on the outer limits of the city and we love it.
Columbia is midway between Kansas City and St. Louis which is nice. Although we have a great mall and other services in the area it is nice to know that you are only 1.5-2 hours away from either city so it is easy to make day trips to see a ball game, visit family/friends or to shop.
If you have any questions or want more info feel free to PM me anytime.
