Daylily seed need cold stratification so the cold frame would work well because you want the seeds to be cold but not frozen. I don't think the soil type matters but, seeds do not need fertilizer. Light won't be important until after the seeds germinate but, too little ventilation will cause mold to grow on the soil surface. The seeds will need moisture to germinate but, but don't keep them too wet.
The seeds can be germinated in damp sand and then transferred to pots when you see roots. I would then use 1 gallon pots with several seeds in each. Or, just start with several seeds in 1 gallon pots.
