Do there exist functions satisfying $f(x+y)=f(x)+f(y)$ that aren't linear?
Yes continuity is enough: You can quickly show that $f(x)=x\cdot f(1)$ for $x\in\mathbb N$, then for $x\in\mathbb Z$ and then for $x\in\mathbb Q$; assuming continuity, this implies validity for all $x\in\mathbb R$.
Any other functions only exist per Axiom of Choice: View $\mathbb R$ as a vector space over $\mathbb Q$ and take any $\mathbb Q$-linear map (which need not be $\mathbb R$-linear).
Also, it is well known that graph $\{(x, f(x):x\in \Bbb R\}$ of every solution $f$ except $f(x)=ax$ is dense in the plane $\Bbb R^2$.