Sounds as though you have all the bases covered, and I can't think of any "sneak" learning disability that would result in refusing to put away toys.

If he's willing to lie about homework, do you think you and his teachers could set up a notebook system? Have him list his homework and ask each teacher to sign that he has everything, or to sign off if he really has nothing. That way, when you ask him about homework, he has to produce concrete evidence about what his assignments are, and you can check to make sure he's actually doing them. Then you have a better way to judge how much he is or is not doing. Of course, this does mean that you have to saddle him up every night, and I don't know to what extent you want to start letting him sink or swim on his own.

It is always deeply frustrating to have a talented child resist work and discipline in this way. You have my sympathies.