Given the kind of syntactic framework Carnie's textbook is written in, the NP/DP Fredrick cannot be an argument of (nor, consequently, receive a Theta role from) consider for two reasons: 1) because that would leave the one-place predicate foolish without the only (= subject) argument it must have, an obligatory Theta role of foolish would not be assigned to/discharged by anything, and the Theta Criterion would be violated, and 2) because that would also require consider to be a three-place predicate, whereas, of course, consider cannot take, and assign Theta roles to, three arguments, as 'arguments' are defined in both logic and P&P linguistics (note that the 'secondary predicate' or 'predicative' that alternative syntactic theories would claim to exist in Wilma considers Fredrick foolish would not be a third 'argument' of consider, but a 'predicate' of Fredrick).
As to why Fredrick must become the subject of the passive clause (iii) Frederick is considered t to be foolish, the reason is that, otherwise, Fredrick would receive no Case and would violate the Case Filter. That, in its turn, follows from the fact that passive participles like considered cannot assign accusative Case (nor any other Case) to their complements. When the complement of consider is a clause, as in (iii), the clause itself causes no Case Filter violation, as clauses do not need Case, but an NP/DP object does need accusative Case. If consider is not made passive, as in (ii), Fredrick can, indeed, receive accusative Case from it, no Case problem arises, and, therefore, Fredrick need not move from its 'deep' position of subject of the infinitive to be foolish. What's more, in such circumstances, Fredrick cannot move at all unless it is into a position in which no Case is assigned, e.g., the position of Topic, as in Fredrick, Wilma considers __ to be rather foolish. However, if consider carries passive inflection, as in (iii), either Fredrick moves into a position in which it can receive some Case from another Case-assigner (here: nominative, from the Tense inflection of the higher verb is) or the sentence violates the Case Filter and is doomed to fail.
Finally, the Case requirement for NP/DPs also explains why * Fredrick to be foolish is considered ___ (by Wilma) cannot be a sentence of English: if the whole infinitival clause is 'moved' and made the subject of is considered by Wilma, the clause itself, of course, raises no Case problem, as explained, but its subject, Fredrick, still remains Case-less and violates the Case Filter, since a non-finite Infl cannot assign Case to an overt subject like Fredrick. If the subject is phonetically null, i.e., PRO, of course, no such Case Filter violation arises - either because null subjects require/support no Case, which seems reasonable, or because non-finite Infl assigns Null Case to PRO, depending on the theory of Case each Chomskian linguist adopts - and the result is well-formed, as in PRO to be foolish is considered __ by Wilma a fatal flaw.