I enjoy gardening and by no means am an expert but know a lot about it. The spider plants can be put in water to ensure a good root system before planting in soil. If they have good roots already, just go ahead and plant in soil. Use a good potting soil mix, not just from the garden.
As for your seedlings being too leggy, do you pinch them as they are growing? Let them develop their 2nd set of leaves before doing anything with them. Once they get start to get some height, pinch them back with your fingers to encourage "bushy" growth. You might have to do this a couple of times throughout the growing season.
As for germination, read your seed packet. Some seeds absolutely need light to germinate where some do not. Some seeds need to be at 70ish degrees to germinate. I would probably keep the light at least 12-18" away from the seed pots. Also use a mister to water. Watering from a watering can, can push the seeds to deep into the soil. Just sprinkle soil over your seeds, you do not need to bury the seeds. Good luck. I have to get on the ball and get mine started. Thanks for the reminder.