Choosing a learning rate for a narrow valley

You minimize $f(x, y) = 6.5x^2 + 12xy + 6.5y^2$ by gradient descent with a fixed step size $\eta$: $v_{k+1} = v_k - \eta\nabla f(v_k)$.

• For which $\eta$ does the method converge from every starting point?
• Which $\eta$ gives the fastest guaranteed convergence, and how many steps does it need to cut the distance to the minimum by a factor of $10^6$?
• A colleague plays safe with $\eta = 0.04$. How many steps does that take in the worst case?
• How many steps would Newton's method need?