I heard that every topological $n$-manifold $M$ is $\mathbb{F}_2$-orientable, but then for $M=\mathbb{R}^2$ is must be $H_2(\mathbb{R}^2;\mathbb{F}_2)\neq 0$?
In lecture we had the lemma: Let $M$ is a connected n-manifold, then: $M$ is $R$-orientable iff $H_n(M;R)\cong R$.
Something must be wrong, I always thought that $H_2(\mathbb{R}^2;\mathbb{F}_2)=0$ because $M$ is contractible. Or is the lemma only true for compact manifolds?
Yes, this is true only for compact manifolds (I assume that your "manifolds" are not allowed to have boundary). In fact, $H_n(M;R)\cong R$ iff $M$ is $R$-orientable and compact. You should be able to find a proof in any text that covers Poincaré duality. For instance, this follows from Theorem 3.26 and Proposition 3.29 of Hatcher's Algebraic Topology (note the hypothesis that $M$ is closed in Theorem 3.26).