First post (dollar): this looks fine. Any specific questions?
(There might be some details worth considering about "for...."-- it's not a typical adjunct, so you might have some of the same complications as with double object constructions. But that's probably fine for now.)
This seems to indicate to me that I need to have another level between the mother AdvP and the daughter AdvP (that is modifying the other Adv). Is this right?
That seems reasonable to me.
Honestly, the specifier is one area that is a real mess in that book. It is first introduced as a place for determiners, then that is revoked several chapters later and the place remains. Eventually it will be the place for subjects in VPs (and other higher V-like phrases). It may also be the location for certain types of modifiers as in measure phrases for PPs ("
[one foot] from the house"), but whether in this case you'd want to put it as specifier or not... probably not. So your second tree works.
Fourth post ("for..."): That works. There are some subtle distinctions in the distribution of "for" and "that", but they can both be considered C heads. (If you want to know more about the distinctions, see Rizzi 1997, "The Fine Structure of the Left Periphery" where C is split into about 5 other projections.)
Disclaimer: it's been a while since I've looked at the book in any detail, so I'm just looking over the work in general, not based on specific details of any particular chapter, etc.
Asking questions is fine! Whether we'll be able to keep up with all of them may vary, but you won't get in trouble for posting them. (Maybe not more than 10 per day or something like that

)