This is basically the same thing you played before. The same answer applies. It does not serve your rational best interest to steal from another, this is the poorest assumption and only serves to underline that the writer of the article makes the same mistake as you do. First rational self-interest understands they have rights, which conversely means that you also have rights. Two with that same understanding and following the philosophy would not try to take what they have not earned from another.
This is also a failure to understand the central theme of the philosophy. What you call "duties towards each other" is simply a recognition of my rights/your rights. You do not steal from another because they have no right to take what they have not earned from you, therefore you do not have a right to take what you did not earn from them. It is rational to act on behalf of their rights if you expect them to act the same.
It is not altruistic to simply recognize that hypocrisy is not a rational action. It is rational to expect others to act with knowledge of your rights, it is a simple logic to act the same towards theirs. It is, in fact, rational to make laws to protect those rights. Hence the illegality of theft.
And this one makes me laugh the most. It is not necessary to list every "negative right". The government has no right to coerce me to change my religion, some thought it necessary to list this right in the Constitution but it doesn't make it necessary to list every right. Observed by Amendment 9...
No, the logical fallacy displayed before doesn't allow for this extension of absurdity.
And this is still a further extension of the fallacy presented from the beginning. The whole article plays off a false assumption which extends further using other logical fallacy (argument to the absurd being the most prevalent, piano shot from a cannon... geebus).
The reality is, almost all of your objection is based on a misunderstanding of the basic premise. It isn't altruism that would cause the rational mind to accept the rights of another, it is simple self-interest. If I expect my rights to be observed I must act the same towards others.
I totally understand the basic premise, Rand thinks that people will do the right thing when they become selfish, but it is not basic human nature to become self involved and thus become more beneficial to others, it is in fact the opposite, i do not misunderstand anything of Rand's philosophy, all I have to do is look to her life. She was very self absorbed, how did her self absorption help others? And in the end, she ended up being the very thing she hated. It seems fitting.