Germination to Seedling

Because we grow hydroponically we have had nothing but problems getting seeds to go anywhere after germination. Obviously we can get them to germinate, then the problems begin. Clones are no trouble at all. We have a cloning machine that holds a 1-1/2" rockwool cubethat we just put a little dab of Clonex on our new clone, put it in the hole in the rockwwol, and 4-5 days later we have roots. We then take the rooted rockwool cube and put it in a larger rockwool cube that will hold that one, and we are all set.
My question is "Can we do something similar with our germinated seeds?" By that I mean can we after the seeds are germinated put them into the rockwool and just leave it in our little 12 hole Cloner until they root just like our clones do?
I hope somebody understands this because I just bought some seeds on this site. OG KUSH, 6 for $92.75! I do not want to screw these up!
My partner and I are sick and tired of growing White Widow, and so are our patients. We need some variety!

Seedlings and clones are a bit different. Clones need to be in high humidity for several days in order to remain healthy enough and retain enough moisture to grow roots. High humidity isn't necessary for seedlings and any such excess moisture could potentially cause them problems (particularly fungal diseases/damping off). Therefore, you don't want them [seedlings] to be covered by any means (dome or otherwise).

I don't believe in 'pre-germinating' seeds by soaking them or putting them in between paper towels. Especially if you'll be growing in rockwool, you can germinate directly in the cubes as they'll retain plenty of moisture (and air) and can simply be re-planted into a larger cube as you described after you see good root formation.

I love the ezcloner with a waterchiller. Not only do I have a 95% succes rate, but healthly beautiful roots, That can be ready to transplant in less time then the rockwool & dome way. The plants r all around healthier too.