Multiplication in elliptical curve cryptography
I understand that $16\cdot 7\equiv 10 \bmod 17$, but where does the $7$ come from? Also, why is $5^{−1}\equiv 7 \bmod 17$ shouldn't that equal zero..?


When working with modular arithmetic, $a^{-1}$ (if it exists) is a number $b$ such that $a\cdot b\equiv 1$. So, because $5\cdot7\equiv 1\bmod{17}$, we say that $5^{-1}\equiv 7\bmod{17}$. Not sure where $0$ came from in your mind as $0$ times anything is $0$.

In step 5, where you are multiplying by $\frac{7}{7}$, the reason they are doing this is to cancel out the bottom since $5\cdot 7\equiv 1$. This step is not necessary, but it does make things simpler, I guess. You can play around with this modular arithmetic calculator to verify that $\frac{16}{5} = 16\cdot 7\equiv 10\bmod{17}$.