Free market capitalists argue that what we have now is not a free market. IP rights wouldn't exist in a free market system, for example.

It's probably best to find another term to describe the market you see working, as 'free market' is what free market capitalists argue for. I don't know what a better term would be, but I'm sure that economists would've thought of one.

Exactly - the government should prevent markets from becoming less free - discouraging monopoly, persecuting fraud, encouraging transparency, taxing (appropriately) externalities, and owning natural monopolies (like infrastructure - dumb pipes owned by the government or a very regulated entity, and used by market competitors). I also think there should be less regulation in certain very regulated markets (banking, food) to encourage experimentation, but those should come with restrictions (e.g. no advertising, limited sales volume, etc.) - I think the world of hedge funds is a good example - they have less restricitons and hence can offer lower costs and better (or less correlated) performance, but they are effectively closed off to the general public (only "knowledgeable" investors can invest).

How do monopolistic entities full of self-interested humans solve problems of externalities?

The economics research on public choice has done a great deal for setting aside the common yet inaccurate assumption that government agents pursue the interests of society. It's not appropriate to compare real world markets to utopian governments. Both are imperfect. The real question is which is superior under realistic assumptions.