First off, the healthcare industry is one of the biggest employers in Missoula. There are two hospitals and all the typical doctors offices, physical therapy, bone and joint, rehab centers. If you are well qualified you can probably get a job. The University of Montana is located here, so there is a fair amount of the liberal element represented in the local population. Still, that's not a reason to disregard living here. The university also results in more culture then lots of western towns (i.e., live theater, park concerts, symphony, etc.) if you are interested in such things.
Weather in the summer is great. Days are very long (light until about 10:00 in mid-June/early-July) and it rarely ever gets too hot. You do have to contend with the smoke in the summer. Missoula is surrounded by mountains and smoke from local fires, as well as fires in Washington, Idaho, and Oregon sometimes settles in here and stays for a while. It some times gets so bad the you can't see across the valley. I can be like living in a campfire.
Winter weather in Missoula is generally mild, but the sun does not come out very much. Some days are very foggy, while most are just overcast.
Fishing and river floating can't be beat. Several famous fly-fishing rivers come right through town such as Clark Fork, Bitterroot, Blackfoot. Rock Creek, a local favorite, is a short drive away. And speaking of Rock Creek, some of the biggest bighorn rams in the lower 48 can be hunted there (not sure how tough it is to draw a tag). Plenty of camping and hiking opportunities abound and if you are willing to drive an hour or so there are some great lakes for fishing, skiing, camping, etc.
As far as elk and mule deer tags, Montana offers residents an over the counter tag for each that is good for any weapon until you fill it (rifle season alone is five weeks long ? not sure how long the archery season is before that). The over the counter tag is good in a very large number of areas, but not everywhere. Tags for some area are by draw only. If you draw a mule deer tag, for example, in a draw only area then you can hunt mule deer only in that area. On the other hand if you get an over the counter tag then it is good anywhere in the state that does not require a draw tag (with some exceptions).
If the housing prices in Missoula have not scared you, then you might consider taking a look at Bozeman. It is more expensive, but the weather is clearer (and typically colder) in the winter. Bozeman has just as good access to hunting and fishing rivers.
Hope this helps.
Mark