|MadSci Network: Astronomy|
The Sun gives us 1370 watts of power per square meter perpendicular to the line of sight (that gets spread out where the surface of the Earth is tilted away from the Sun, so that only the point nearest the Sun gets the full 1370 watts). The Earth stays at an approximately constant temperature because it's radiating the same amount of power back into space that it receives from the Sun. It's radiating from all over the Earth, while it's getting energy only on the day side; it's also radiating the energy back into space at infrared wavelengths while the incoming energy is in visible form.
One way to guess at how fast things would cool off is to estimate the amount of thermal energy in the atmosphere and hydrosphere and divide that by the energy loss rate. If I do this in the very crudest way, then the estimate is that it'd take a couple of centuries for the temperature to fall by about 100 degrees C. Certainly things cool off faster than that at first (how fast does the temperature fall after sunset?) but the whole hydrosphere would cool rather more slowly. In fact, since ice floats an serves as an insulating blanket, it would take a much longer time than this, probably, for the Earth's oceans to freeze solid, and in any case there would be small volumes of liquid water around te geothermal vents on the ocean bottom for a very long time.
In terms of equipment and satellites, you're right, most satellites orbiting Earth are solar-panel powered, so they would become non-functional very quickly. Other equipment on Earth would last longer, though we'd be in serious trouble as the ecosystem collapsed and agriculture failed when the light for photosynthesis went away.
Try the links in the MadSci Library for more information on Astronomy.