Rice’s Theorem for Turing machine with fixed output

So I was supposed to prove with the help of Rice’s Theorem whether the language: $ L_{5} = \{w \in \{0,1\}^{*}|\forall x \in \{0,1\}^{*}, M_{w}(w) =x\}$ is decidable.

First of all: I don’t understand, why we can use Rice’s Theorem in the first place: If I would chose two Turingmachines $ M_{w}$ and $ M_{w’}$ with $ w \neq w’$ then I would get $ M_{w}(w) = M_{w’}(w) = x$ . But this would lead to $ w’$ not being in $ L_{5}$ and $ w \in L_{5}$ . Or am I misunderstanding something?

Second: The solution says, that the Language $ L_{5}$ is decidable as $ L_{5} = \emptyset$ because the output is clearly determined with a fixed input. But why is that so? I thought that $ L_{5}$ is not empty because there are TM which output x on their own input and there are some which do not.