Many things behave differently in high dimensional space
Along each direction for a unit cube, we have $2$ boundaries. To be less than $0.01$ from a bounday in a $d$-dimensional unit cube, it is not inside the cube of side length $1-2\times 0.001$ sharing the same centroid as the original cube.  

The corresponding volume for that is $(1-2\times 0.01)^d$ where $d$ is the coresponding dimension. 

$$\left(1-\left( 1-2\times 0.001\right)^2\right) \times 100 \% \approx 0.3996\%$$

$$\left(1-\left( 1-2\times 0.001\right)^{10,000}\right) \times 100 \% \approx 99.999999798\%$$

When we first computed the answer for $2$ dimension, we might carelessly conclude that most points are not closed to the boundary. The moral of the story is be careful not to generalize what we believe without proof to higher dimension.

Your second question is also known as Hypercube line picking of which its computation is non-trivial. We know that 

$$\frac13 \sqrt{d} \le \Delta(d) \le \sqrt{\frac{d}{18}\left( 1+2\sqrt{1-\frac{3}{5d}}\right)}$$