Let $g \in C(\mathbb{R}^n, \mathbb{R}^n)$ with $K > 0$ such that $|g(x) - x| \leq K$.
- Is there a function g without a fixed point?
- Is there a non-injective function g?
- Is g always surjective?
ad 1.: for $n=1$, I think $g(x)=x+1$ might work: $$g(x) - x = 1 =:K > 0 $$ but $g(x)$ obviously has no fixed point $g(x^*)=x^*$.
ad 2.: I think $g$ is always injective because the $\ker(g)$ is trivial.
ad 3.: No idea.
I will discuss the last question. Without loss of generality, you can show that any $g$ with the stated properties has a zero: the equation $g(x)=y$ is reduced to $g(x)=0$ by translation.
Define $f(x)=x-g(x)$, so that $g(x)=0$ amounts to $f(x)=x$. In other words, you are looking for a fixed point of $f$. But your assumption says that $|f(x)| \leq K$ for every $x \in \mathbb{R}^n$, so that $f \colon \overline{B(0,K)} \to \overline{B(0,K)}$. You can now use Brouwer's theorem to conclude.