My first guess is that there's a certain subpopulation for whom, if you treat their problems casually, joke about them, etc., then they'll get offended or hurt and cry while pointing a finger at you (perhaps literally), and thereby bring a bunch of social opprobrium upon you; and the acquaintances don't know enough to distinguish you from that subpopulation, and therefore treat you as though you might be one of them.  It's a "safe" strategy they're following; the cost to them if they're wrong is relatively small, compared to the cost to them making the opposite error.  (And perhaps they've done that all their life and don't have any practice joking about rough problems.)

(Take all guesses I make, about people I don't understand, with a big grain of salt.)

You might enjoy Gilbert and Sullivan's satirical light opera "Patience".  In it, there's a poet that all the women of a village have fallen deeply in love with, except one woman, named Patience, who doesn't like him; and unfortunately the poet seems interested in Patience and not in any of the other women, who are therefore miserable and moping about it.  Patience has never been in love, doesn't understand it, naively asks questions about it, and is puzzled by the answers.  It thus serves as a vehicle to poke fun at various contemporary notions of what love is supposed to be.

MAIDENS.
Twenty love-sick maidens we,
Love-sick all against our will.
Twenty years hence we shall be
Twenty love-sick maidens still!
...
ANGELA Love feeds on hope, they say, or love will die;
MAIDENS Ah, misery!
ANGELA Yet my love lives, although no hope have I!
MAIDENS Ah, misery!
...
MAIDENS All our love is all for one, Yet that love he heedeth not,
He is coy and cares for none,
Sad and sorry is our lot!
Ah, misery!
...
PATIENCE.
I cannot tell what this love may be
That cometh to all but not to me.
It cannot be kind as they'd imply, 
Or why do these ladies sigh?
It cannot be joy and rapture deep,
Or why do these gentle ladies weep?
It cannot be blissful as 'tis said,
Or why are their eyes so wondrous red?
...

ANGELA. Ah, Patience, if you have never loved, you have never known true happiness! (All sigh.)
PATIENCE. But the truly happy always seem to have so much on their minds. The truly happy never seem quite well.
JANE. There is a transcendentality of delirium – an acute accentuation of supremest ecstasy – which the earthy might easily mistake for indigestion. But it is not indigestion – it is æsthetic transfiguration!
...
PATIENCE.
If love is a thorn, they show no wit
Who foolishly hug and foster it.
If love is a weed, how simple they
Who gather it, day by day!
If love is a nettle that makes you smart,
Then why do you wear it next your heart?
And if it be none of these, say I,
Ah, why do you sit and sob and sigh?

(The biggest missing factor in Patience's model is probably the fact that the maidens' love is unrequited.  Though this is complicated by the fact that some people do enjoy fantasizing about not-necessarily-requited love, at least some of the time.)

Later, Patience gets the idea that love must be selfless... And therefore, it's improper for her to love someone who has lots of good qualities, because that would benefit her; instead she should marry an awful person, because living with them is absolutely unselfish.  So she agrees to marry that poet, Bunthorne, who is vain, posturing, moody, petty, etc.  But then Bunthorne promises to reform himself into a good man.  Patience is initially delighted by this, but then realizes the implications.

PATIENCE. Oh, Reginald, I’m so happy! Oh, dear, dear Reginald, I cannot express the joy I feel at this change. It will no longer be a duty to love you, but a pleasure — a rapture — an ecstasy!
BUN. My darling! [embracing her]
PATIENCE. But — oh, horror! [recoiling from him]
BUN. What’s the matter?
PATIENCE. Is it quite certain that you have absolutely reformed — that you are henceforth a perfect being — utterly free from defect of any kind?
BUN. It is quite certain. I have sworn it.
PATIENCE. Then I never can be yours! [crossing to R.C.]
BUN. Why not?
PATIENCE. Love, to be pure, must be absolutely unselfish, and there can be nothing unselfish in loving so perfect a being as you have now become!
BUN. But, stop a bit. I don’t want to change — I’ll relapse — I’ll be as I was —

I would say that there is a place, in proper love relationships, for a thing that might at first glance resemble "unselfishness".  But that thing is less "assigning zero value to your own happiness / well-being / etc." and more "assigning a similar value to your partner's utility as to your own", so that e.g. if something costs yourself 10 utils and benefits her 20 utils, you'll do it (and in a healthy relationship, lots of things like this happen in both directions and it's net positive for both).  But it's pretty fair to say that general cultural transmission doesn't make things like this clear.

Absolutely.  For a quick model of why you get multiplicative results:

• Intelligence—raw intellectual horsepower—might be considered a force-multiplier, whereby you produce more intellectual work per hour spent working.
• Motivation (combined with say, health) determines how much time you spend working.  We could quantify it as hours per week.
• Taste determines the quality of the project you choose to work on.  We might quantify it as "the expected value, per unit of intellectual work, of the project".

Then you literally multiply those three quantities together and it's the expected value per week of your intellectual work.  My mentor says that these are the three most important traits that determine the best scientists.

Also:

• Knowing the importance of the advantages people have makes you better able to judge how well people are likely to do, which lets you make better decisions when e.g. investing in someone's company or deciding who to hire for an important role (or marry).
• Also orients you towards figuring out the difficult-to-see advantages people must have (or must lack), given the level of success that they've achieved and their visible advantages.
• If you're in a position to influence what advantages people end up with—for example, by affecting the genes your children get, or what education and training you arrange for them—then you can estimate how much each of those is worth and prioritize accordingly.

One of the things you probably notice is that having some advantages tends to make other advantages more valuable.  Certainly career-wise, several of those things are like, "If you're doing badly on this dimension, then you may be unable to work at all, or be limited to far less valuable roles".  For example, if one person's crippling anxiety takes them from 'law firm partner making $1 million' to 'law analyst making $200k', and another person's crippling anxiety takes them from 'bank teller making $50k' to 'unemployed', then, well, from a utilitarian perspective, fixing one person's problems is worth a lot more than the other's.  That is probably already acted upon today—the former law partner is more able to pay for therapy/whatever—but it could inform people who are deciding how to allocate scarce resources to young people, such as the student versions of the potential law partner and bank teller.

(Of course, the people who originally wrote about "privilege" would probably disagree in the strongest possible terms with the conclusions of the above lines of reasoning.)

On the angle of demonstrating that you can learn the material and the skills and generally proving your math mettle: Can you study the books, do a sampling of the problems in the back of each chapter until you think you've mastered it, and then take the tests directly, without being signed up for a class?  Maybe find old exams, perhaps from other institutions (surely someone somewhere has published an exam on each subject)?  Or, for that matter, print out copies of old Putnam contests, set a timer, and see how well you do?

As someone who never entered college in the first place, I consider it a prosocial thing to make college degrees less correlated with competence.  Don't add to the tragedy of that commons!

I grew up knowing "privilege" to mean a special right that was granted to you based on your job/role (like free food for those who work at some restaurants) or perhaps granted by authorities due to good behavior (and would be taken away for misusing it).  Note also that the word itself, "privi"-"lege", means "private law": a law that applies to you in particular.

Rights and laws are social things, defined by how others treat you.  To say that your physical health is a privilege therefore seems like either a category error, or a claim that other people treated you better in a way that gave you your better physical health, which then raises questions like "What made you deserve that treatment?" or perhaps "Is it really because of how other people treated you, or other reasons like genetics or having made healthier life choices?".  The latter may then lead to "Yeah, but you grew up being taught better and/or in a situation where healthy choices were more affordable, which are probably partly caused by wealth and are privilege", both of which might be counter-argued in the specific person's case or in general, and so on.  Social justice arguments ensue.

"Advantage" seems like a more neutral term, one that doesn't inherently imply fairness-laden claims about how you got it.  I would recommend it.

The first, second, and third considerations in such a study would be ruling out other directions of causality:

• Does having a sore throat make people hum less?
• Does being healthy make people happy, and therefore inclined to hum more?
• Does humming correlate with being in a choir, which may also cause respiratory diseases to be spread more often (in the absence of precautions)?
• Does living in close quarters with lots of other people make them likely to stop you from humming, and also make them more likely to pass diseases onto you?
  • Does having this happen early in life give you a stronger immune system or at least plenty of immunity to common diseases?
• Do certain cultures have attitudes towards humming and also attitudes to hygiene that are causally relevant?
• ...

I would be extremely, extremely skeptical of any study on the subject other than a randomized intervention.

I had heard, 15+ years ago (visiting neuroscience exhibits somewhere), about experiments involving people who, due to brain damage, can no longer form new memories.  And Wiki agrees with what I remember hearing about some cases: that, although they couldn't remember any new events, if you had them practice a skill, they would get good at it, and on future occasions would remain good at it (despite not remembering having learned it).  I'd heard that an exception was that they couldn't get good at Tetris.

Takeaway: "Memory" is not a uniform thing, and things that disrupt memory don't necessarily disrupt all of it.  So beware of that in any such testing.  In fact, given some technique that purportedly blocks memory formation, "Exactly what memory does it block?" is a primary thing to investigate.

One class of variance in cognitive test results is probably, effectively, pseudorandomness.

Suppose there's a problem, and there are five plausible solutions you might try, two of which will work.  Then your performance is effectively determined by the order in which you end up trying solutions.  And if your skills and knowledge don't give you a strong reason to prefer any of them, then it'll presumably be determined in a pseudorandom way: whichever comes to mind first.  Maybe being cold subconsciously reminds you of when you were thinking about stuff connected to Solution B, or discourages you from thinking about Solution C.  Thus, you could get a reliably reproducible result that temperature affects your performance on a given test, even if it has no "real" effect on how well your mind works and wouldn't generalize to other tests.

This should be addressable by simply taking more, different, cognitive tests to confirm any effect you think you've found.

Pithy sayings are lossily compressed.