First off, ignore Killyjoy. Don't ever cut a shotgun barrel, that's fucking stupid.
Now then, from personal experience I've learned that a good home defense gun doesn't make a good hunting/clay/trap/target shooting gun. I bought a Mossberg 500 12 gauge some years ago with that in mind, and quickly learned it really can't do everything.
For home defense, you will want something with a short barrel (18.5" is the shortest you will find for legal reasons), and for the wife you will want 20 gauge. As for keeping it from the kids, get some hooks to hang above a door or up on a wall, high enough your wife can reach but not the kids. Also, don't keep anything in the chamber and keep the safety on. That's three steps the kids would have to get past in order to do any damage.
As for a home defense gun. I suggest the Mossberg 500. The reason is that the safety is more intuitive, especially in a high-stress situation like a break-in. Also, I don't know if this is true with all of them, but mine has a rather heavy trigger. This means that you are far less likely to accidentally pull the trigger, you have to WANT to pull it. The downside to the trigger is that it means that when doing ranged shooting, you are more likely to jerk off target, which is why I suggest this only for home defense.
For all of the other shooting, you can't go wrong with a Remington 870, but for cool points I suggest finding a nice over/under. 12 gauge with at least a 24" barrel is a better choice here, far more options in ammo and more likely to hit a small target.
Just remember that no one thing is best at everything. If you HAVE to compromise, I'd probably lean towards an 870 with a short barrel and long barrel in whatever gauge your wife is comfortable with, but compromising sucks.