Does the temperature of space affect the weather on Earth?

Not in the least, except that by being colder than the surface of the Earth, it provides a 'sink' for the thermal energy and radiation to go somewhere. If space were HOTTER than the Earth, radiation would flow FROM space TO the Earth's surface and not the other way around. Although the Earth receives radiation from the Sun, it also emits infrared radiation into space to try to cool off. If space were warmer than say 270 K, Earth would not be able to cool off in this way and would continue to heat up until it reached a new equilibrium temperature hotter than the surrounding space.