The question How are the schema, memory and knowledge changed after having an ill-structured problem solved? currently has 2 downvotes. I know this is not personal, and it's good to moderate the site, and the downvoters don't need to comment, but still I can't help but wonder why. Where is it unclear? Let's say the therapy bit is unclear, but why not just skip it and see the question as a whole? I think the question as a whole is perfectly clear, and one should not dowvote it just because of a unclear example.
How are the schema, memory and knowledge changed after having an ill-structured problem solved?
What do you expect an answer to contain? I can give a trite answer: you remember how to solve a specific trick question. Generally these are not helpful for advancing knowledge much. With your marriage example, you might learn (or be reminded) that to marry is slightly polysemic:
b : to give in marriage [e.g.] married his daughter to his partner's son
c : to take as spouse : WED [e.g.] married the girl next door
d : to perform the ceremony of marriage for [e.g.] a priest will marry them
Your examples are ill-structured puzzle (trick) questions. Not the more general kind of ill-structured problems:
Ill-structured problems are ones students face routinely in everyday life. They include important social, political, economic, and scientific problems (Simon, 1973). In order to resemble situations in the real world, ill-structured problems have unclear goals and incomplete information (Voss, 1988). [...]
How does a designer create ill-structured problems?
Design a complicated problem that we face in everyday life. Ill-structured problems should come from a real-life situation in which there is no obvious right answer. Problems should be authentic and relevant to students (Howard, McGee, Shin, & Shia, 2001). Ill-structured problems should include vaguely defined goals. The information available to the decision maker should be incomplete or ambiguous (Wood, 1993). Problems should make it unclear which concepts, rules, and principles are necessary for the solution.
Design a problem including multiple solutions and perspectives. Ill-structured problems must allow alternative solutions instead of one correct answer (Meacham & Emont, 1989). Additionally, ill-structured problems should allow students to pursue different procedures for solving the problem. These various procedures will come from allowing different perspectives based on students’ perceptions and interpretations of the nature of the problem.
And a proposed definition of trick question includes these elements:
Intention: Items are considered to be tricky if the test constructor intended the item to be confusing or misleading. Thus, it is not the content of the item per se but rather the intention of the item writer.
Trivial Content: Items are considered to be tricky if the content of the item is trivial and unimportant but that the trivial point is the focus of the correct answer. [...]
Stem Includes Noise Material: Items with long stems, that contain information totally unrelated to the required skills to answer the item, are considered to be tricky. This component is similar to what is referred to in the literature as window dressing.
Your (new) hat rack example is also probably more of the latter kind with the useless top beams highlighted [noise material] and trivial content (most would know that hanging a hat from a C-clamp is possible).
Of course whether something is trivial or not depends on the context. If you use the marriage question to open a lecture on polysemy, it would probably be a welcome (and non-trivial|) example in that context. If you use it as a job interview question for a programmer...
The Cog Psy textbook you cite (which contains both examples you gave) has nothing to say about the knowledge gained; all it talks about is how such questions are used in research on insight.
Alas, if given in classroom, what students will generally learn in response to trick questions is not to answer them:
But teachers know there is often no such answer to many of their questions; are sure an incorrect one will be given; they expect at times that an answer not be given; and at other times they themselves supply the answer. These are the rhetorical, leading, loaded, trick, trap, test or tough questions put 'to make students really think'. No matter what students think, the answer to such questions is necessarily wrong. Hence students will give cautious, minimal responses if they give any at all. 'After a certain number of bad experiences, many students learn not to volunteer answers to riddles, "come-ons", or invitations'. [cited to: LABOV, W. The Study of Nonstandard English (National Council of Teachers of English, Urbana, Illinois, 1970)]
In general, such puzzle questions are used (e.g. in interviews) as a proxy for IQ testing.
Why do organizations use these types of interview questions? There appears to be two primary reasons. First, puzzle questions originated in the uncertain high-technology industry, where employees are generally expected to be logical, creative, and innovative (Poundstone, 2003a). Thus, the conventional wisdom among interviewers who use puzzle questions is that they are a useful measure of an applicant’s cognitive capabilities such as critical thinking, creativity, intuition, flexibility, and ability to reason (Munk & Oliver, 1997). Second, puzzle interviews are seen as an opportunity to assess the applicant’s reactions to a novel, uncomfortable situation. Some employers feel that the widespread availability of information about conventional interviews, through the Internet and other sources, allows candidates to prepare their responses in advance of the formal interview (Poundstone, 2003a). Thus, the puzzle interview is seen as an opportunity to assess the candidate in an unrehearsed setting.
which doesn't mean this is the best way to test that...
Results showed that puzzle interview performance was correlated with scores on the Wonderlic (r = .45, p < .01), providing some support to the hypothesis that puzzle interviews measure cognitive skills. Applicant reactions results, however, were mixed. Participants who performed well on the puzzle interview, and those who perceived that they performed well, reported that the process was more procedurally fair than those who performed poorly (or perceived that they performed poorly). However, there were no significant correlations between applicant reaction measures of perceived predictive validity or face validity and either perceived or actual performance on the puzzle interview. The authors did indicate, however, that participants’ reactions to the puzzle interview were generally negative. Descriptive statistics showed that the means of two of the three applicant reactions measures (perceived predictive validity and face validity) were all below the midpoints of the maximum scale values.
The quote I gave you above shows that something else can be learned: a bad impression of the interviewer in a setting where the questions are expected to be fair and on specific skills.
So what is learned substantially depends on the context.