According to Wikipedia (my books are in storage): "A simple graph with n vertices (n ≥ 3) is Hamiltonian if each vertex has degree n / 2 or greater (Dirac 1952)"

Each vertex of G has degree 6 >= 10/2, so G is Hamiltonian.

Each vertex of G-u has degree 5 or 6 >= 9/2, so G-u is Hamiltonian.

Each vertex of G-u-v has degree 4, 5 or 6 >= 8/2, so G-u-v is Hamiltonian.