Is λx. a valid Lamda Calculus abstraction?
No, $\lambda x.$ is not a syntactically valid expression. By the definition of the syntax of lamda terms,


  $\lambda x.M$ is a term if $x$ is a variable and $M$ is a term.


You need a term $M$ to abstract over, and there are no empty terms. (This also means that there can be no application that reduces to "nothing". Any lambda term must consist of at least an atom (= a variable or a constant)). So $\lambda x.$ is not a term.