1. The wiki should be more than adequate for this
2. Same as above
3. This is a little more complicated since the update has change a few things with combat but the basics are:
a. Go to the military screen and create a new squad, choose metal armor uniform if you want infantry, archer otherwise
b. Select people to join the squad (you do this by -> on the squad and then browsing the list of names and pressing enter when people want are highlighted)
c. Set up a barracks with (optional) weapons/armor depot (you do this by placing a weapon/armor rack or chest)
d. Set up a schedule for your military with the squad screen (different from the military screen!)
i. This is its own guide by itself, I recommend looking at the wiki for advanced setups (patrols, station's etc.) but you really only need training
ii. The squad should already be setup to train all year, just activate the squad schedule from the first screen and you are good to go
e. When danger comes, go to the squad screen again, select a squad, select (k)ill, select enemy (or group of enemies) and watch the carnage
Some caveats: untrained units should not wear full suits of mail as that will now significant slow them down. Don't send raw recruits to do anything except train because they will get murdered (or run). I believe training has now been fixed (somewhat) so that squads of 10 can train just as well as squads of 2. It takes about a year or two to get an untrained recruit to legendary status just by training. Danger rooms will take about 2-3 days, though they no longer train fighter, they still train weapon/armor and dodge skill, so they are still extremely effective.
As for the other questions, I feel should give some advice (though again, the wiki contains all the info you need):
1. Mine carts are currently not very useful except for very specific tasks (I like using them to shuttle ore (dropped through 200 level mine shafts) to my magma forges. As for how they work, basically you need rails and you need a cart. The rails can be built (i.e constructed) or carved out by miners. (note that the latter can only create straight rails, you have to construct curves) The rails need start and end points, these points can then be connected to stockpiles and set on timers to shuttle items in a timely fashion.
2. Quantum stockpiles are a bug and unless you are suffering from extreme lag or just can't be bothered, I don't recommend using them. They do make stockpiling incredibly efficient (to the point where it becomes cheaty, in my opinion). The idea is pretty simple, when dwarves dump something (instead of stockpiling it) the dump square can hold an infinite number of items. This is the basic quantum sp. The more advanced version uses the minecart trackstop connected to a stockpile. You can look at the wiki for that.
Seriously though, most of this stuff is on the wiki. I'm surprised you didn't find it newbie friendly (I find its more useful than any player-made tutorial)