Consider the infinite strip, $[0,1]\times\mathbb{R}$ with $[0,y]$ identified with $[1,y]$ for all $y\in\mathbb{R}$, and with the Riemannian metric $g=dxdx+dydy$. Consider $X=f(x)\partial_x +\partial_y$. The Laplacian is $\Delta X =f''(x) \partial_x$.

For the first question, let $f=-x$, so that the line $x=0$ is a limit cycle and $\Delta X=0$. For the second question, let $f=x^2$, so that $x=0$ is again a periodic orbit, but now, everywhere on the curve, $\Delta X=2\partial_x$ is orthogonal to the tangent $\partial_y$.

Consider the infinite strip, $[0,1]\times\mathbb{R}$ with $[0,y]$ identified with $[1,y]$ for all $y\in\mathbb{R}$, and with the Riemannian metric $g=dxdx+dydy$. Consider $X=\partial_x +f(y)\partial_y$. The Laplacian is $\Delta X =f''(y) \partial_y$.

For the first question, let $f=-y$, so that the line $y=0$ is a limit cycle and $\Delta X=0$. For the second question, let $f=y^2$, so that $y=0$ is again a periodic orbit, but now, everywhere on the curve, $\Delta X=2\partial_y$ is orthogonal to the tangent $\partial_x$.