> My interpretation, leaning on the verbose side, is "There is a house which has an interior which has an accusative role (meaning, it has the ability to be the direct object of an action) which has an action (it has an instantiation, or, it has a subject-object framework which it can be described in)." 

Sounds good to me! However the house is a noun, right? If it was a verb, it would be "something that actions is related to something that accusatives that is related to something that interiors that is related to something that houses". Because not only houses house, a company can also house people. Or a second problem: the girl kicks a cat which is an object, hence a noun; she doesn't kick an action; kicking is the action. 
> I realize you said "bear in mind that the girl does not kick the house", but it seems like you can't make that distinction without a slightly more complex semantics. 

True, you need a language theory to explain it, and there can be more than one. To me the depth of the parse tree seems to suggest something. 

> I *think* I have several solutions to this, but I haven't finished messing around with the options yet. I promise to make a detailed post of some sort...