Whether there is some compact manifold $M$ such that $f:M\rightarrow M$ is continuous $\Rightarrow f$ has a fix point.
I mean that any continuous map from $M$ to $M$ must have a fixed point.
I really just guess it. And know nothing about whether it is right or not. Maybe Klein bottle is an example, but I don't know how to prove it .
Yes, such manifolds exist, even without boundary. For example, $\mathbb{CP}^{2n}$ can be shown to be such a manifold for all $n$ by the Lefschetz fixed point theorem.
With boundary there are simpler examples, such as the disks $D^n$, by the Brouwer fixed point theorem.