There are a lot of answers relating to these questions on this website, I encourage you to look around for some more detailed responses to specific aspects, as well as maybe suggesting the answers to this question for further reading:
http://www.anarchy101.org/1435/who-should-i-read-up-on.
This is the sort of argument/discussion I find myself in most often with non-anarchist friends and enemies (though enemies tend to be far less logical and tend to resort to misconstruing and insulting anarchy in general). So I do put some thought in to it. Whereas I used to spend lots of time trying to come up with a comprehensive outline or illustration, more recently I've mostly stopped, because the reality is we don't really know. That is because (as I say again and again and again), it wil be different to different groups and individuals.
As far as food and meeting basic needs, people have provided those things for far longer without the state and capitalism than they have with. that doens't necessarily mean everyone will be anarchist hunter-gatherers, or that our lifestyle (any lifestyle) is sustainable with the number of people alive now, but that is irellevant to answering that question. We are a pretty resourceful species, and people will do what they need to do to survive, which, combined with nature doing what it needs to do to regain equilibrium will probably deal with population pressures (for example, I assume people will die of starvation - not necessarily some massive die off, but it's probably gonna happen. If people claim you are a genocidist for saying such, they are liberals or full of shit. People will either create autonomous networks to transport food from area to area, or they will provide it locally for themselves.)
If no one wants to do necessary tasks, then those tasks won't get done. A lot of "necessary tasks" are not actually necessary to human survival, such as, waste management. People have been making waste since before humans were humans, but managing it is an issue of civilization, not humanity. If people want to live in cities, they will need to figure out that and many other tasks, or the cities will look a lot more like those of an earlier age (or maybe the favelas and slums of the global south - which, incidentally, provide some great examples of mutual aid in the vaccum of state authority, in addition to providing some great examples of what happens if capitalism exists withouth the state). If no one wants to do it, perhaps Bob Black (in his piece "The Abolition of Work," available for reading here:
http://theanarchistlibrary.org/library/bob-black-the-abolition-of-work) has the best idea: "Small children who notoriously relish wallowing in filth could be organized in 'Little Hordes' to clean toilets and empty the garbage, with medals awarded to the outstanding."
So contemporary anarchists (this is mostly repeating some of the stuff I said in my answer to the first link I listed) include: Colin Ward, Peter Gelderloos, Uri Gordon, and probably a whole lot of anarcho-communists I haven't read.
If this sounds flip, I (sort of) apologize. It is an honest question, and one that people are truly concerned about, but it is also a question loaded with assumptions about shared values, levels of technology, population concentration, and a whole lot more. This is why many here distinguish between anarch/ism/ and anarch/y/.
