Are there any other circumstances? (Is the wood burner mooded? Is he on break?
Is he a dirty bastard vampire? Is he doing something else?) You can try toggling the labor on and off a few times. A screenshot or two might help determine what is wrong, as well. Also, I'd recommend getting
Dwarf Therapist very, very strongly.
2. An additional benefit of this 'feature' is that you can set up a statue garden with the confines of your fortress to prevent your dwarves from developing cave adaption.
3. It's... complex. I'll start by assuming that you've already figured out how to assign a militia commander and put dwarves into a squad.
I'll try to break it down simply:
1. Ensure that you have the equipment you're assigning to your dwarves. This includes every piece of armor and weapon (according to whatever type and material specifications you selected), as well as a backpack and a flask
or a waterskin. If you're using marksdwarves, they will also need quivers and ammuntion, which are a whole 'nother can of worms. Anyhow, assign the desired combination of equipment with: (m) > [select squad] > (e). You can also create uniform templates and assign them to a squad as a whole. Keep in mind that your dwarves will not collect their equipment until the squad is called up or assigned a training schedule. Of course, for this, you'll need to have made or bought weapons and armor; if you don't have something, the dwarves can't equip it.
2. Assign a schedule. From the core military menu, it is (s). Everything is labelled; from here, you can select the squad you want to schedule for, and create, copy, paste and delete orders for each month.
The wiki can help you substantially on this.3. Make sure you have a barracks. This is a room created from using the (q) selector on a bed, armor rack, or weapon rack. With beds, you will need to assign the room as a bedroom, and then toggle the selector to make it a barracks. From the barracks menu, you can assign squads to do various things in it: train, sleep, store their equipment when it isn't needed.
4. (m) > (a) [set your squad to active duty/training] (Not your CIV, though.)
That is just the bare-bones of it. Once you have a squad actively training, you can order them to move to a location, patrol, or attack a creature or sentient from the (s) menu.