$\displaystyle \gamma = \frac{1}{\sqrt{1  \frac{v^{2}}{c^{2}}}}$
When you travel at 0.999 speed of light then to you what 3 seconds has passed is actually 67.2 seconds to an observer on earth if you are in a spaceship that is traveling fast.

Close. I think its 67.1
Also 1 meter to you on the spaceship would be observed as 0.45 meters on earth. This way time dilation works and length contraction also.

No, I think the contraction is observed by you on the spaceship, not by the person on the earth.
Gamma is "cooked up" so that all observers see the speed of light as the same. So one observer will measure light travelling across a distance x in time t as c = x/t and another observer travelling at velocity v will see light travelling across a distance x/gamma and taking time t/gamma so to keep the speed of light the same we would need to calculate the length contraction of 1 meter as 1/gamma(0.999).
Time dilation is a bit confusing because it refers to the time you would see viewing a clock on earth with a very good telescope from your spaceship. It's not the time you measure directly like length contraction. That's why we divide by gamma for both x and t to change distance and time on earth to results we would see from spaceship.