1

"My philosophy, in essence, is the concept of man as a heroic being, with his own happiness as the moral purpose of his life, with productive achievement as his noblest activity, and reason as his only absolute."

As she herself confesses, this is a concept, her concept. I, for one, do not agree with this premise, and so everything flowing from it fails. Separate to this, she rambles and shambles on about moral purpose and productive achievement, let alone employing the term noble—I mean, really?
Let me paraphrase, These are things I like and think are important because I like them and think they are important. Sorry, but #FAIL.

One should not seek happiness that would in turn in infringe on other people's life or rights.

First, we need to ask why happiness is a goal followed very quickly by is happiness achievable—or perhaps more relevantly, is persistent happiness achievable. Current psychology says no, so this may be a fool's errand, tilting at windmills.
Without having an answer to this first line of questioning, the second part is not particularly meaningful.

2

Well, morality and philosophy are concepts, so I don't see where that gets us.
Also, why is happiness a goal? She argues that one should br concerned with their own interest. One's own life is the most important and you must be a bit selfish in order to preserve your life and secure your own happiness. Which is worked for, she argues.

3

Rand's philosophy was sophomoric and incoherent.  BUT, she is a good polemicist.  And adolescents who feel overwhelmed by the obligations of human Eusociality like being told that they do not owe the world a payback for the investment in raising them.  It is the attraction of being told that being immoral is just fine.

4

While I did read her "Virtues of Selfishness" and I could grasp where she was coming from, her philosophy can only take one so far. Ultimately, I don't think that selfishness will help a person to grow. Charity, while difficult to employ in a perceptual environment, is most rewarding when mastered. That doesn't mean that everyone is deserving and you should just go giving of yourself, all willy nilly! You still need to use the power of your own discretion. Nothing in life is easy!

5

Indeed, so the situation might be better framed by asking—assuming that  that happiness is attainable, sustainable, and measurable—should we optimise the happiness of one over the optimisation of happiness for the many?
Ursula K Le Guin tangentially tackles this subject in her short story The Ones Who Walk Away from Omelas (PDF, 4 pages), a city where one person suffers in order for society as a whole no to. Of course, this is a critique of Utilitarianism.