I've found the multiple class thing going on, too, or splits between classes and other dwarves sparring. Suits me fine - less standing around waiting that way. Still, , they all seem to level up in various things -- though they won't level up in, say, shield use until I've made them shields. They often seem to do dodging classes -- moreso than other skills, but seem to gain the other skills (like melee skills) just fine as well. For the record, I put the most experienced dwarf in the squad in the leadership position when possible, and try to make sure that I mix in some experienced dwarves with fresh squads since I don't think they can do classes unless someone in the squad knows the skill (though how much they need exactly I don't know).
And yeah, they do seem to switch into civilian status every so often. If you ever give the squad orders (like to station somewhere) they'll revert back right away, though sadly they do seem to stow their equipment when they go civ and then have to go back and get it.
For what it's worth, despite training them every month of the year, my soldiers aren't unhappy - other than long patrol duty (if they're stationed), they don't get a lot of bad thoughts from training all the time. They don't have a lot of chances for good thoughts either though, so I try to keep them on 'sleep in room at will' when practical, and give them nice furniture (this is a standard for me in general, my dwarves are always admiring their beds and cabinets), as well as nice food and drink stockpiles nearby and making sure they all have waterskins/flasks and backpacks on hand for them I need to station them someplace for awhile. Seems to work okay for my purposes, though I'm pretty much a newbie myself.