Let $a_n=\cos(a_{n-1}), L=[a_1,a_2,...,a_n,...].$ Is there an $a_0$ such that $L$ is dense in$[-1,1]?$

Ok, if you study recursive sequences, then you probably heard about "Lamere Ladder".

According to Lamere Ladder for $\cos(x)$ (which is also a contraction because of MVT), this function has a fixed (stationary) point. So, regardless of $a_0$, $a_1$ will end in between $[-1, 1]$ and from there on, the sequence $\{a_n\}$ will tend to the fixed point of $\cos(x)$. Which makes $L$ a converging sequence, so $L$ can't be dense, because the only point satisfying ii) (in your question) is its limit.

On another note, Kronecker's approximation theorem is quite an useful tool too. For example $\{n+ m \cdot 2 \cdot \pi \space | \space m,n \in \mathbb{Z} \}$ is dense on $\mathbb{R}$ and $cos(x)$ is a continuous function, making $\{cos(n)\}_{n \in \mathbb{Z}}$ dense on $[-1,1]$.