Is it possible and desirable to directly train the abs' strength?
Of course it's possible.
Now, the abdominals' primary function is to stabilize the spine. We see this because of its cross-hatched design. There is the rectus abdmonius, transverse abdominus, and obliques. These are oriented in different directions and so produce force in different vectors. Since the primary purpose of the abdominals is stabilizing the trunk, they can be strengthened by exercises where you must do this, like squats, deadlifts, presses, and Olympic lifts. Sine these exercises use a lot of weight, they are very effective at strengthening the abs.
Training the abdominals for athletics is more involved, since you also want to use them to produce directional force rather than use them only isometrically. So you'll want to do some special exercises in line with the needs of the sport.
Mike Robertson of IFAST in Indiana has written extensively on abdominal training. His material is a good resource.