Will a dwarf with some broken bones get better even if he is not diagnosed?
None of my dwarfs have any skill in diagnose and I had an accidental cave in.
A dwarf can still perform a diagnosis when they are unskilled. It can, however, lead to
disaster fun. Or nothing at all. Anyway, simply choose a chief doctor and enable the Diagnose labor for them (and/or any other dwarves you want performing said labor).
Not sure if your injured dwarf will heal their broken bones on their own. At the very least it would take a very long time without the aid of casts and splints. Even longer if they "heal slowly." I've had a dwarf in traction for
six years because they heal slowly.
1. How do I train up social skills when I have an industrious fortress (I want my mayor to become a great speaker/social dwarf. Also, he is in the military but I don't think this counts [That is in one of my other threads]) Now I what I want to know is how to get another dwarf inactive and make them chat in my 2x2 meeting area, they go to their own bedrooms most of the time.
Disable all their labors. With nothing to do, your chosen social trainer will hang out in the meeting area and throw parties. They might also get a little upset about not having any work, though. Alternatively, super-specialize a dwarf for a labor that doesn't need constant attention (or the extra dwarf you'd be adding), like farming. Or milling. If you only have one millstone but two or three millers, one of them is always sure to be inactive. If you have add an extra planter/grower in vanilla - or even if your grower is just skilled enough - they'll have plenty of down-time. My smiths are almost never doing anything once my military's equipped. Disable your military labors, too, and your off-duty squads MIGHT decide to chat it up instead of do individual combat drills. If your future mayor is in the military, this could be part of his problem. But if he's doing any sparring or combat drills he could be chatting with his fellow soldiers.
Double-alternatively, look into the Higher Learning mod, which adds a library you can use to train many skills, including social ones. Would require genning a new world, though.
You could also try using burrows to encourage interaction between specific dwarves. Make small burrow, assign two dwarves to it (and no other burrows), watch them sit there and wonder why they aren't allowed to leave the room.