Assuming you want to capture animals (groundhogs, horses, bears, dreaded Rhesus Macauqes) as opposed to vermin.
Build a cage trap where the animals roam (usually next to an outdoor food pile works well) with build, Traps, cage. You will need a mechanism and a cage. Mechanics skill is used to place the trap.
You should probably designate an animal stockpile (p, a) where the animal will be taken after it's caught.
After this, just wait for the animal to be caught.
Note: the capture live land animal option at the kennels catches vermin not animals.
Hint: placing your cage traps randomly can take too long to capture anything.
One way to increase the chances of an animal walking into your traps is to channel one long, 1-tile wide, dry moat(d,h) and remove all ramps(d,z) except one in the middle of the moat. Then you surround the single ramp with traps.
Once you've captured an animal, even if you're going to butcher it you probably want to get it trained at least a little, to avoid it breaking free on the way to the butcher shop. To train, make sure you have at least one dwarf with 'animal training' labor enabled, then find the animal in the status tab (z) and designate it for training. Butchering is also handled through the same status menu.
Yes, children can die from any method just like any other dwarf. You can't control children at all, but unless it a baby, which this doesn't seem to be, they will feed themselves and work farms. She'll mature if she lives to be thirteen years old, and migrants will still come, so this fort may have some hope!
Best Answer
The traps article in the wiki mentions that you can capture nearly anything in a cage trap if the cage trap square also has Giant Cave Spider webbing on it. Otherwise, friendly creatures don't set off traps (but wild animals do). This method would require you to get a Giant Cave Spider to web your traps, which is a serious undertaking.
There is another possible option. The same article mentions that your dwarves can become trapped if they lose consciousness while on the same tile as a cage trap. It is possible, although I have not tested it, that the same would apply to merchants. Unfortunately it is much harder to make merchants lose consciousness.
On further thought, it is possible that the merchants no longer count as friendly after you attack them, but any traps they have seen while friendly might not work. If the merchants are human then a human diplomat sort of effect may have occured, which would mean that any traps previously seen by the diplomat (or perhaps merchants) would not be effective.
To reiterate; the options I see are three. One, use a Giant Cave Spider. Two figure out how to knock the merchant unconscious while he is on the right tile. This may or may not work. Three make sure the merchant chases you over traps he has never seen before. Again, this may or may not work.
Option one is the most likely to work. Do note, however, that there are downsides to messing with merchants too much. Read the caravan article, especially the point about loyalty cascade as it can kill your fortress. Also if prevented from leaving they can go insane and attack your dwarves.
Good luck.