The volume of an $n$-dimensional simplex of unit edge length is $$V(n) = \frac{\sqrt{n+1}}{n! 2^{n/2}} \;,$$ so at least $\lceil 1/V(n) \rceil$ such simplices are needed to cover the unit $n$-cube.

. What is an upperbound on the number of unit simplices needed to cover a unit $n$-cube?Q1

I suspect that the $1/V(n)$ lowerbound is weak; perhaps there is no $c>0$ such that $c/V(n)$ is an upperbound?

For $n{=}2$, $1/V(2)=4/\sqrt{3} \approx 2.3$, but $4$ equilateral triangles are needed. For $n{=}3$, $1/V(3)=6\sqrt{2} \approx 8.5$; I don't know how many tetrahedra are needed to cover the cube.

. How many unit tetrahedra are needed to cover a unit cube?Q2