Would it be proper to have an Objectivist hero in a story cry?

Objectivists are humans, so of course they routinely feel emotion like anyone would (barring some medical problem). You can see that reflected throughout Rand's stories because she's writing about humans. Exactly how someone reacts-to or acts-on his emotions is another matter.

What Objectivism urges (naturally) is being objective about the proper relationship between cognition and emotion: these are both essential in a healthy human life, and it is crucial to not make the mistake of trying to substitute one for the other, denying either it's critical role. Err one direction and you'll suffer the effects of repression; err the other direction and you'll suffer the effects of emotionalism.

Emotion is no sin in Objectivist ethics, but rather "automatic results of man’s value judgments integrated by his subconscious."

From the Playboy interview:

An emotion is an automatic response, an automatic effect of man’s value premises. An effect, not a cause. There is no necessary clash, no dichotomy between man’s reason and his emotions—provided he observes their proper relationship. A rational man knows—or makes it a point to discover—the source of his emotions, the basic premises from which they come; if his premises are wrong, he corrects them. He never acts on emotions for which he cannot account, the meaning of which he does not understand.

Some emotions can be irrational if they are premised on faulty values or inconsistent with reality. It's up to us, through introspection, to explore, define, and validate our emotions.

A new update to the original question asks (bold emphasis added):

...what about the person who isn't as strong and intelligent as, say, the valedictorian in school, the millionaire who made a fortune in the stock market, the genius who cures cancer, or the contractor who designs skyscrapers? Certainly that person has goals, ambitions, and dreams. What if he doesn't have the capacity to acquire what he most desires with honor? What would be the point of going on if day in and day out a person experiences the same disappointments, knowing he can't have what he wants because he's not smart enough to hold a job which would pay his dream house, as one example?

Objectivism can offer the following guidance to a person of lesser ability as described above:

• Everyone needs to decide for himself how he wants to live his own life (or let others tell him what to do and perhaps help him to do it, whose instructions he will dutifully follow).

• If he believes others somehow must help him to live, whether they want to or not, and he advocates political policies designed to force others to help, they will be morally entitled to resist such attacks and to defend themselves. The individuals of lesser ability who supported such policies will end up worse off than they could have been if they had not supported the malicious efforts to shackle the top producers.

• If an individual of lesser ability doesn't have the talent and/or motivation to become a top producer, he can still be a lesser producer. He can still be self-supporting and generally very decent morally. If he is serious about living his own life to the best of his ability, he will adjust his sights to whatever is attainable by him, as far as he can judge it. If he nevertheless strives for too much and fails to achieve it, he deserves to suffer such loss as the price he must pay for his refusal to be realistic about himself and his opportunities.

If one cannot be a top producer, for whatever reason, what reason does one have to continue living at all? Objectivism would say that he still has the capacity to go on living on a level that is commensurate with his own ability and effort, and it can be a highly fulfilling and rewarding life for anyone who sets realistic goals and works consistently over time to achieve them. Does he want to live or not? If life is what he seeks, he can still seek it -- and probably achieve it.

There is a scene in Atlas Shrugged that encapsulates the issue in economic terms. Dagny is being given a tour of the Valley by Galt, and she meets Andrew Stockton. He explains how he "ruined" a competitor in the Valley, and how the "ruined" competitor is better off because of it, making more money working for Stockton than he made without him, and doing more of the kind of work he loves. Dagny asks:

Then somebody could put you out of business, too?

[Stockton:] Sure. Any time. I know one man who could and probably will, when he gets here. But boy! -- I'd work for him as a cinder sweeper. He'd blast through this valley like a rocket. He'd triple everybody's production.

[Dagny:] Who's that?

[Stockton:] Hank Rearden.

(p. 724hc, slightly less than half way through Chapter I in Part III. Refer also to the topic of "Pyramid of Ability" in The Ayn Rand Lexicon.)

The updated question concludes:

I don't think I'm a smart person. I get terrible grades in math and science.

The questioner has often demonstrated in his postings, and continues to demonstrate, that he is quite intelligent in certain areas, if not math and science. He has said that he is prevented from pursuing the kind of career he wants because he can't yet afford it and can't think of a practical plan to be able to afford it. He lives at home while attending college, with considerable financial assistance from his parents, who are ordering him to pursue a school curriculum that he hates; but he can't defy them without risking being kicked out of their house, and can't afford to live on his own. (He is welcome to correct or update any of this if it's no longer accurate.) As I've said before, he needs a plan (aka an "exit strategy"). If he can't think of a plan, then I guess he'll have to keep doing whatever his parents tell him to do, if he wants to eat.

Objectivism can guide anyone, at any level, who wants to live (on whatever level of ability and opportunity). Objectivism isn't only for the top producers, although they certainly merit full recognition for who they are and what they do, especially the great benefits that everyone else receives from their productive achievements ("pyramid of ability"). If one can't be a top producer oneself, one can still be perfectly happy to work for a producer who needs competent, supportive workers. The fact that the top producers deserve the top moral appraisal does not in any way preclude lesser producers from deserving a full moral sanction of their own, as well, commensurate with what they have earned by their own ability and initiative. Again, it comes down to the question: how strongly does one want to live? And: what is the most appropriate way for man to live?