They won't stay in the cooler area longer than they need to. In your laundry room you could use a couple of 60-100 watt bulbs adjusted to give 95F at chick level. Make sure which ever way you do it that the box is large enough that they can get away from the heat if they need to but you will find that inside the temperature will remain fairly constant once you have it set right. Outside there will be major temperature fluctuations which could cause a problem, especially at night. When they are fullly feathered you can give them the run of the stall.
Wood shavings are fine for litter once they are a few weeks old.
Initially you should use an absorbent towel, like bounty to keep them clean and dry. If you have any other questions, just ask. This is a very helpful board. Yes, you should either cover to stop owls or any other predator from getting in. The other option would be to enclose your brooder.