Onion sets will bolt very early, as they are actually onion that was grown from seed the previous year.There are not as many varieties available as with seeds.
Onion seeds are one type of seed that do not store well. I learned that the hard way one year. I planted 2 yr old left over seed, and had less than 10% germinate.
Once the majority of the seeds have germinated, it is safe to take the plastic dome lid off.
This is only used to ensure that the soil stays damp for germination.
As your onions starts grow, they may get long and scraggly. It is perfectly ok to give them a "haircut" to neaten them up. I trim them down to 2-3" high.
Winter sowing is a great option for starting onion seed if you don't have a lot of room indoors. Plant the containers with the seed, set it outside, simple as that.