L = { | M is a TM and halts on every input after 5 steps exactly } - is L decidable?
It is NOT NECESSARY to read the whole input for a Turing Machine to halt.

Imagine a Turing Machine M that receives any input such that it reads anything and moves right exactly 5 times. On fifth step it reaches the accept state and halts.

M belongs to L.

Contrary to what I was thinking before, L is indeed decidable.

Consider all &Sigma; $^5$ configurations of the first 5 cells of the tape. For each of these configurations execute 5 steps. If TM is nondeterministic execute 5 steps for all possible paths. If TM does not halt for some configuration (or execution when TM is nondeterministic) reject. Otherwise, accept.

As  &Sigma; $^5$ is finite, L is decidable.