I'm new to this so take what I say with a grain of salt, hopefully more experienced growers will chime in, but yes, I would put them in dirt. Seeds grow straight out of dirt in nature, so it's not like you're adding a challenge to the roots that they're not evolved to handle.
I'd take the bottom half of a plastic soda bottle and make little individual domes for each of the solo cups you put them in and keep misting the seedlings and get your lights closer to them.