this was my fourth year growing yacon (grew nearly 70 plants as my only crop to market this year, not done digging them all yet!)...I'll take a stab at this. warning: many words.
yes, put only one in a space as small as 3x3. plant as soon as danger of frost is gone in spring (or a little earlier and cover well for mild frosts). a hard frost or freeze will kill them back to the ground. in my experience yacon likes good thick mulch and fairly regular feeding (I like to build a sheet-compost style mulch around them once the plants are in the ground, with alternating layers of leafmold and chopped greens [comfrey, for preference], and do occasional extra feedings up until sometime in about september [we get our killing frosts in november, usually]). decent nutrition and avoiding dry out is key.
regarding harvest/care: for me, harvest happens anytime after a decent hard frost kills the tops. be careful when digging, because the tubers are super brittle when being dug. I usually find it best to loosen the soil around the whole plant at a diameter of about 3 feet, and then pry the whole rootball out with a well-placed deep shovel and someone lifting on the stems, cut at 6 or 8 inches long for handles (I've gotten pretty good at prying and pulling by myself, but it's like awkward weightlifting/yoga, and I'd recommend it being a 2-person job) once the rootball is out, I gently separate out tubers and the crown. where I am, it makes sense to bring the crowns in for storage in a cool dark place (the basement, in damp mix or soil) while winter happens. they're usually starting to grow again in late winter/early spring, so then I start separating the pieces and potting them up as weather allows. it's nice for the plants to have decent-sized leaves before they go in the ground.
I've experimented with storage/care of the harvested tubers a bit. stored in the basement, under damp soil, they keep just like when they were dug (probably with slight chemical differences), with thin skin, firm/hard feel, and not much sweetness in the flavor. if they aren't covered down there, they begin to wrinkle. on the other hand, once you clean the tubers and bring them into a room-temperature environment, the real curing and sweetening starts, and the skin gets a little thicker and darkens toward purple (they can dry out a bit and get a little wrinkly, but frequently the thicker skin minimizes this), and they get sweeter and sweeter. I like them after they've been curing in a spot that gets some sun for a week or so. refrigerating tubers at any point seems to lead to a slimy blackening that is most unappealing. there may be some value in curing them all together in a waxed cardboard box, as opposed to a loose pile or single layer, to minimize water-loss. the ones I've collected in 1/2 bushel boxes for selling seem to be curing without wrinkling up much.
that's enough for now. I'll try to get a couple picture to add to this when I get a chance. gardenRN jeff may have insights from his first year growing them, too...