Mark wrote:

The vast majority of the seeds sprouted quick, and are growing madly-
most of them have their cotyledon leaves & stems almost to the top of
the plastic greenhouse top- they look kind of spindly right now. No
true leaves have shown up yet.

Is this a problem, or is it common for the cotyledon leaves to shoot
up
really high and quick? I read that lack of light will cause leggy
plants, but I'm not short of light with my setup, am I?

you are definitely short of light. the other pieces of advice are
pretty good, and they will improve your light efficiency and prevent
damping off, but they won't solve your main problem, which is too
little light. You can evaluate things yourself. In full sun, with the
sun overhead you get about 600W/m2 of light. With the sun at 45
degrees, you get 420W/m2. Each 72-cell tray is about 0.18m2, so you
should be coming in with close to 80W/tray. Those fluorescents are
fairly efficient, but they are not 100% efficient, so make that
100W/tray. I found that the only thing that helps is a sunny window,
and taking them out to full sun as soon as feasible.