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While vertical factory farms would be pretty nifty, how useful would they really be?

• Let's consider a medium-sized produce farm parcel as 640 acres (one square mile) and should be able to feed around 1000 people (order of magnitude estimate). To get that amount of space, Manhattan would need to sacrifice a square block and build to 100 stories. This seems to fail the cost-benefit test unless they can produce orders of magnitude more food/acre than traditional farming.
• How many do you need? Let's look at Manhattan (population 1.5M). Assuming that a vertical farm is 10 times as productive as a traditional farm, you need around 150 of them to support the island. This would be around 1.5 square miles of building footprint (about 15% of the island).
• Livestock produce less food/acre than plants unless you truck in all of the food. Using it for animal "crops" (as the article suggested) seems to be a worse idea than just growing fresh green stuff and trucking the meat in.
• "All of the water in the entire complex would be recycled" isn't possible. A bunch of that water is being turned into food and shipped outside the farm. Remeber the Lunar Revolution? Presumably, they are expecting to process the urban sewage and use that for water and fertilizer.
• No diseases or parasites? BioSphere2 didn't manage to stay that isolated. Besides, where do they expect to get the seeds/grafts, fertilizers, water, and staff? Do they expect the entire facility to operate as a clean-room?

Ultimately, if you want to reduce transport costs (money, fuel, etc.) the people need to be closer to the food production. This seems like an idea better suited to lower-density, urban sprawl (where you can grab relatively large areas without consuming a large percentage of the available space) rather than in the middle of compact urban areas.

"How many and what type of plants would it take to convert the carbon dioxide exhaled by the astronauts and convert it to enough oxygen for them to survive?"

Not sure, but I believe there was an experiment in Arizona which used a helluvalotta plants, and found that it wasn't enough. It was only a first attempt, though, and should give some idea what would be required for a permanet moon base --where you could also mine water and oxygen, so you needn't rely entirely on self sustaining ecosystem until you get it tweaked just right. You'd need extensive use of grow lights up there, with 350 hours or so of darkness in a lunar day.

Hmm, the Biosphere in Arizona anyone?

Does anybody else think this sounds kind of like a repeat of the Biosphere experiments? Except with less living space and fewer windows...

Biosphere is a (formerly) sealed facility outside of Tucson Arizona that was used to see if we could create a closed habatat that could support human life for an extended period with no outside supply source. There's museum there now, that's basically just a glorified greenhouse. They had a few teams of people who were going to live in it for a while, but it started to leak, and there was a build-up of undesireable gasses that they couldn't control, so they were forced to end the experiment early. Supposedly they learned from their mistakes but it was too expensive to try again.

A one-way trip would be an interesting idea if there were a reasonable chance of survival on the other end. The problem is that we humans have a very primitive understanding of what it takes to make a self-sustaining ecosystem, particularly one complex and robust enough to support humans. Full-ecosystem projects like Biosphere 2 are not only spectacular failures, they are so far beyond our understanding that most scientists don't even believe they yield scientifically valuable information.

If you believe that building self-sustaining colonies away from Earth should be a long-term goal of humanity (as I do), then we need to start with research here on Earth focused on understanding and learning to engineer these kinds of self-sustaining ecosystems. We have to be modest enough to realize there are many baby steps between our current understanding and any hope of self-sufficiency away from Earth.