Decidability of Turing machines that never move their heads past any input string
The halting problem, $\mathsf{HALT}$ reduces to $\overline{L_2}$.

Given a TM $T$ and input $w$, create a new TM $N$ that on any input of length $n$, simulates $T$ on input $w$ for $n$ steps and then stops except that if $T$ ever halts before $n$ steps, $N$ will move its head to the right forever.

There is a gap in the above reduction. When $N$ simulates $T$ on $w$ for $n$ steps, it may move out of the input when it moves left (we assume that input is put to the right of the origin, the initial position of $N$'s head, inclusively). This gap can be resolved by the classic trick of "translating the tape". When the simulation is about the move to the left of origin, let $N$ translate the current of the tape one cell to the right. Then $N$ goes to the origin, as if it were the cell to the left of the origin. In this way, we will make sure $N$ will never move out of the input string as long as the simulated $T$ on $w$ does not halt. (To enable $N$ to recognize the origin, it should always mark the origin with a "compound" symbol that also tells its original symbol. For example, if the original symbol at the origin is $A$, $N$ should change it to $A_o$, a symbol that is not $A$ but points to $A$. "Compound" symbols are also used when $N$ translates the content of the tape.)



Since $\mathsf{HALT}$ is not decidable, $\overline{L_2}$ is not decidable. So, $L_2$ is not decidable.