Increase thinking capability through relativity 
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*As Einsteins relativity suggests that everything slows down along with the speed of light. So, if we were to live in space would our thinking capability (essentially speed) increase or decrease (due to the increase in light's speed)?

*Also, can we change (increase) our thinking speed by staying as still as possible?
 A: Yes, if you lived in space far from the Solar System you would think slightly faster than you do on Earth.
However the connection to velocity is a red herring. As Alfred says in his comment, you're at rest in your own rest frame so you're already thinking as fast as you can. Re your Q2: the fastest speeds regularly experienced by humans (airplane flights?) make so little difference that they can be ignored, so I wouldn't bother making the effort to sit still.
However there is a small time dilation due to (a) being in the Solar System's gravity well and (b) being on the Earth that is whizzing round the Sun. There is a calculation of the time dilation in the question Time Dilation Effects from simply being on a spinning planet orbiting a star in a rotating galaxy in an expanding universe.. The result is:
$$ \frac{t}{t_0} = 0.999999984 $$
So someone floating in space far from the Sun and at rest with respect to it would be thinking about 0.000002% faster than you. Hardly worth the effort really.
