Childfree

Let's start with this one:

  if everyone thought like you, the world would disappear 

But everyone doesn't, and everyone never will, so it is a pointless comment. What if everyone thought like person X and wanted to be a brain surgeon? We would have 8 billion brain surgeons and no farmers, business people, etc. and we would all die (via reductio ad absurdum). If on the other hand, a person is encouraging everyone to be childless (as in that is the ideal life for which everyone should strive) that this comment would be appropriate and a good response. As it is now, we can call it a strawman fallacy due to what the responder is implying (i.e., the childless person wants everyone to hold their position).

  there are people who would be happy to have children, but they can't

This argument appears to go as follows:

p1. Some people can't do X.
p2. You can do X.
C. Therefore, you should do X.

Note the should. This indicates opinion. We can call this a simple non sequitur since it doesn't logically follow that we should do something simply because someone else can't do it.

it's not natural not to have a child, reproduction is the basis of life

appeal to nature

This all remind me of Kant's Categorical Imperative.

Some actions become bad only when enough people do them but before that point they are good.  Kant is in trouble himself here because he did not have any children, so if everyone acted like Kant did as a universal law the human race is doomed.

I'll preface by saying that if someone says they don't want to have kids...there's no problem with that...pro-child people are weird.

IF everyone thought like you, the world would disappear 

This is  Absurd Extrapolation . The same logic is used to argue against homosexuality ("if everyone were gay, the world would disappear") which ignores the fact that you can't transfer sexuality (and given the increase in availability of child-conception technology, is a bogus point). If someone chooses not to have children, that's a decision that only affects one person; it doesn't 'transfer' onto other people (who are free to have kids if they want). So not 'everyone' is going to make the same choice, thus we don't have this issue.

See Dr Bo's comment on this, too.

there are people who would be happy to have children, but they can't

So? "What does that have to do with me?" one would be tempted to ask.

it's not natural not to have a child, reproduction is the basis of life

Faeces is also natural, but I don't want much to do with it other than getting it out of my system after a hefty meal. Natural =/= good.

The rest of the arguments are generic appeals to emotion, and the penultimate one in particular ("you are selfish") is ironic, given that forcing your desires onto someone else is textbook selfishness.

" it's not natural not to have a child, reproduction is the basis of life "

My children were born in '76 & '78. They aren't my biological children, but became a part of my life in '83. I married their mother in '87, and we chose not to have more children.

 if everyone thought like you, the world would disappear

This statement itself is logical. If everyone chose not to reproduce, the species would indeed be doomed. However, making the statement commits the strawman fallacy. It takes your specific argument (I don't need to reproduce) and turns it into an absurd absolute (all people don't need to reproduce) in order to make it easier to attack.

One good reply is to recognize the self-aggrandizing nature of the hypothetical and be self-effacing. For instance, you could say: "I'm not that influential." Or: "I can barely get one other person to think like me, let alone the whole world."

If you are looking for a more logical reply, you could also use the counter-argument the late economist Gordon Tullock used when he told people he didn't vote and they said:  "What if everyone thought like you and didn't vote? We wouldn't have a democracy!" He would reply: "If nobody else voted, then I would vote ... The fewer other people vote, the more likely I am to vote." (Then watch their heads explode as they try to calculate a equally rational counter-argument.)

there are people who would be happy to have children, but they can't 

you will die alone

your life will be unfulfilled

These are all examples of the broad appeal to emotion fallacy (sentimens superior). Put another way, these are pathos (emotional) appeals, which are one type of rhetoric (persuasion). You can attempt to counter them with another type of rhetoric (i.e. a logos appeal that cites/explains the fallacious logic of the argument), but you may be better off countering with an equally potent pathos appeal.

For instance, you could talk about overpopulation and its negative effects on the planet, or explain that many married couples you know ended up divorced because of the stress their relationship experienced when they were raising children. Etc.

In other words: Fight pathos with pathos. ;-)