If you have a $100 budget, you could just purchase a Hova-bator. Or, you could buy an Incukit, which is the heat source, thermostat, and fan all in one, and then just mount it in a cooler.
Yes, but as I said, you'll need a thermostat of some kind to control the lightbulb or you're going to have temperature swings and probably won't get any chicks to hatch.
You can make an incubator with a cooler and a light bulb, but you also need to have a thermostat of some kind to turn the bulb on and off to control the temperature. They sell digital thermostats to control them, or you can use a wafer thermostat or wire the bulb to a water heater thermostat...