Just an FYI, I grew carrots and beets last fall for the first time (very successful crop). I sowed seeds too early, too, and the carrots took forever to come up. The beets came up quickly when they did like the temps. Keep the soil moist and THIN, THIN, THIN, if you don't want stunted carrots and smallish beets.
I grew Bull's Blood and Chioggia in a small bed filled with rich organic compost, Black Kow composted manure, and coffee grinds. The medium was VERY, VERY loose. Only real problems I had were 1) I planted the beets too shallow so they grew almost on top of the soil, and 2) my homemade compost attracted every pillbug that ya'll did not have and they started munching on my beet green tops (which are TO DIE FOR!).
This year, I'll sow my seeds Labor day weekend. Since I discovered how wonderful the beet greens are, I'm growing strictly for the tops this time. Will be using Sluggo Plus to handle the pillbugs and snails.
Oh, and these are what I consider to be high moisture content veggies, so they'll take water -- try not to let 'em dry out, but don't drown 'em either. Good, fast drainage is key, and coco coir should work beautifully.