Assuming the "10" after your name is your growing zone, it doesn't matter, for now, what soil you use, unless you can grow them in air-conditioning with full sun! They're considered a "winter season" crop, although when they say that, they're talking about growing near the frost dates (not literally in winter.) Found a site that tells you more included when, how and what to grow. (Carrots go from round, like golf balls, to the size found in supermarkets, and container gardeners generally need the shorter varieties.) As for soil, since you're growing in a container anyway, seems regular potting soil works fine, but, unless you want some very strange shaped carrots (which are fun, if you have kids, since they might come out with arms and legs), keep anything hard - stones or pebbles - out of it. Check out the link below for info from real pros. This is my first year growing them, and mine are growing slower then the seed package said they were supposed to grow, and we only have four plants quite a few less then planned, even for my small container! Here is a link that might be useful: Growing Carrots - carrot museum (no kidding LOL)

Last year I grew one of those purple varieties here in North Florida along with a mix of other colors in one of the big black concrete mixing tubs you can get from Lowe's (sometimes called "All-Purpose Super Tub)". The tubs are 2 foot by 3 foot by 8 inches deep and hold 4 cubic feet. I put one drain hole in the center and supported it, with other tubs, on landscape timbers placed on concrete blocks. I think the soil mix I used was perlite, compost, and cypress sawdust (I usually use pine bark fines instead of the cypress but I wanted a smoother texture for the carrots). Ozmocote was the main fertilizer and the tubs were all drip irrigated so they never had any water stress. Worked fine. Only thing I'd change for carrots would be to replace the cypress sawdust with coarse builder's sand. You can probably use just about any tub but unless the carrots are very long varieties needing depth I'd stick with a shallow tray-like tub at least 6 inches deep. I've seen long carrots in shallow tubs bend over in the middle when the roots reach the bottom and that makes them hard to harvest whole. So look on the seed package and make sure the container depth just barely exceeds the length of the carrot. That way you will minimize the amount of potting mix you need. The smaller concrete mixing tub (26 inches by 20 inches by 6 inches) ought to work if you want a smaller area. Or any of the shallow storage containers you can find at Wall-Mart and such. Didn't think much of the purple carrot - it turned out rather knobby compared to the others in the bin and it wasn't nearly as sweet as the hybrid orange ones in my carrot mix. But it was unusual!

Hmmm.... just past the cotyledon stage, they shouldn't really need that much nutrition. I'd guess more of a sunlight or soil texture issue. Is the soil fine and fluffy or dense, heavy? While I doubt it's lack of nitrogen at this stage, no harm in feeding them lightly. (I'd sprinkle liberally with coffee grounds and water in).

I've stored carrots a couple times in lightly moist sand in 5 gallon buckets and they were in great shape through at least mid February when we used the last of them. I just alternated a layer of sand, carrots, sand, carrots etc. It's best to keep the bucket in the garage where it stays in the 40s but above freezing. This year I just coined, blanched, and froze all but about 3 lbs that we stored in the fridge and were used before Christmas.