An eigenvector keeps its direction under the matrix but may be stretched, shrunk or flipped.

True

True. $Av = \lambda v$ preserves the direction and scales by $\lambda$, so the vector may lengthen, shorten or reverse; only $\lambda = 1$ leaves it untouched. That preserved direction is what makes eigenvectors the axes along which repeated application behaves simply, the basis of PageRank and PCA.