Yes from late feburary, I start mine off on a sunny windowsil in seed trays in gritty compost. Prick them out when they have their first true leaves (you'll know what Cosmos leaves should look like) pot on to 5cm pots and off they go outside to the propagator, then pot them on again to 10cm pots until they're big enough to grow in the ground around April.
You can sow Cosmos directly in to the ground as well, although I'm not much of a fan of that as the seedlings are tasty to slugs, snails and mice. If you want to do this though it should work, just sow more seeds than you think is necessary.
I grow pure white ones, seashells and the maroon ones you've mentioned this way.