I have a 2' x 4' domed skylight on a pitched roof, facing southeast. If I build a shelf so the seedlings sit a foot under this dome, will it be enough light to prevent leggy plants?
It's too early to start them off where you are - wait till earliest mid March through to mid April. If you start them now, even if they grow sufficiently well, which isn't likely because light levels are a bit low, they'll be hanging around inside as larger plants till you're able to get them outside. You need overnight temperatures of no less than 50 deg F to put them outside later on. The seeds also need heat - 75-85 def F is about right, so you may need to put them on top of a radiator or something if you don't have a propagator. I imagine the solution you propose to give them light will work, provided no direct sunlight gets on the seedlings, and the time to give them good light is once they're germinated.
Bamboo is correct! You have to provide much better grow lights, great draining soil, proper nutrients for your tomatoes. To take them to another environment outside or wherever...would entail an hour the first day, two the next and so on...you can purchase grow lights for starts on up to fruiting/seeding. What zone do you live? If these plants are to be taken outside for maturation, I would do what Bamboo says, wait until you have less than 3 weeks before the last possible freeze. Then begin to acclimate them to their new environment.