It's explained in 2 ways (the 2003 version and the first movie). Splinter watches Hamato Yoshi do his training, and mimics his moves. That's how in the movie Shredder recognizes the Turtles' fighting style.
In many anime I have noticed that a good number of teens live alone while their parents "work overseas." I can believe it when a parent is really busy and thus hardly ever home, but I mean, come on! There isn't any problem with this? How are bills paid? Shouldn't they be living with a guardian? Wouldn't the school or police notice and do something? I don't care if their neighbor checks on them, I just don't believe any decent parent would do this.