Answer: You can still start your seedlings in the unheated shed if you provide bottom heat for your seedlings. You can build a hotbed, or use soil heat cables, or special seedling heat mats. The latter two products are available from garden centers. Ideally, the air temperature shouldn't fall below 50F at night, so if you can work out some sort of heating arrangement, that would be best.
Some long-blooming, sun-loving annuals you can grow from seed are: zinnias, marigolds, alyssum, nasturtium, California poppies, cosmos, celosia, salvia, cleome, portulaca, Ice plant, nicotiana. Whew! Quite a list to start with! You can start these as late as 6 weeks before your final expected frost date, so you have plenty of time yet.
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