I used small mason jars for soaking because I could close them up ans hope the cat or anyone else wouldn't knock them off but they at least have a better chance than being knocked over and emptied. I've seen folks use shot glasses so I really think it's at your discretion.
Here's a post showing what I used:
Alafornia's 1st Time Multistrain Medical Grow
I didn't cover them until daylight (did this at night) and I don't think it matters much unless you let the tap roots come out while in the cup. I wouldn't wait that long. After they soak up enough water to sink they're good to go in the soil. You'll have to wait 4-7 days, give or take, for them to surface once you've planted. Only go down about a 1/4 to 1/2 inch or so with the seeds in the soil.