Question 1 asks if forward Euler is a single step method

If you remember from the lecture, this is true

In fact, forward Euler is the most basic of all single-step methods

It takes exactly one step, and no interpolation is done

Question 2 asks whether forward Euler and trapezoidal are members of the family of Runge-Kutta methods

This is true

Forward Euler is a first order Runge-Kutta method, and trapezoidal is a second order Runge-Kutta method

So this is true

Question 3 asks what symplectic ODE solvers are good for

Symplectic ODE solvers are good for systems that are conservative

That is, systems where you want to conserve energy, or area in the more abstract case

With these systems, failure to use a symplectic ODE solver can sometimes inject friction, essentially

You can imagine, with a pendulum for example, in the non-dissipative case, if your ODE solver is injecting a little bit of error every step size, every time you take a step, this could effectively act as a numerical friction term

This loss of energy fails to conserve energy, and symplectic ODE solvers are developed specifically to keep the energy in the system

Or the area, depending on the system youre working with