If there were no gravity in space, the space shuttle would not be able to orbit the Earth, the moon would not orbit the Earth, and the Earth would not orbit the Sun. They aren't actually weightless, they are still being pulled down by gravity but they and the space shuttle are in a constant state of freefall around the Earth. So they seem to be weightless as a result of the falling - just as you would seem weightless if you were in an elevator when the cable broke. G and g are the same thing. It is not a universal constant.
Gravity is a force, little g is an acceleration.
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