There's no singular school of thought, but yes libertarians almost universally favor government prevention of contagion. It's also not unusual for us to favor some environmental regulations (though by no means approved by all of us!).

Markets are only one facet of a healthy functioning society. You still need government and it's courts and law enforcement. Civil courts deal with the negative externalities (pollution, contagion, etc), not markets. There are of course myriad instances of humans banding together on their own to solve tragedy of the commons issues, without the courts or regulation, but of course in the US anti-trust laws would prevent such coordination as "cartel like behavior".

But let's assume the "ultimate extreme libertarian society", in which even the air we breathe has been privatized. The air shareholders would still need to use the courts to enforce their ownership rights against polluters (a factory, for example). Without the authority of the court to back them up, the air shareholder would never be able to actually force the factory to pay for the right to pollute. The market helps ensure reasonably efficient allocation of air rights, but the courts (government) are still required to actually make the system function.

And finally, to answer your first question: Why would you need an FDA to restrict antibiotic use in agriculture? Normal courts, cops, and laws can't do that?