 Here is the Python code using the capytaine library to perform the tasks as described:

```python
import capytaine as cp
import numpy as np

# Set up logging
cp.set_log_level(cp.LogLevel.INFO)
cp.set_log_format("%(asctime)s - %(levelname)s - %(message)s")

# Create sphere mesh
radius = 1.0
center = np.array([0.0, 0.0, 0.0])
resolution = 64
mesh = cp.mesh_sphere(radius, center, resolution)
body = cp.Body(mesh, cp.DofType.TRANSLATION)

# Extract immersed part of the mesh
immersed_mesh = cp.extract_immersed_mesh(mesh)

# Set up BEMSolver
solver = cp.BEMSolver()

# Define diffraction problem
wave_direction = np.array([np.cos(np.pi/4), np.sin(np.pi/4), 0.0])
omega = 2.0 * np.pi
diffraction_problem = cp.DiffractionProblem(immersed_mesh, wave_direction, omega)
diffraction_solution = solver.solve(diffraction_problem)

# Define radiation problem
radiating_dof = 0  # Assuming translation in x-direction
radiation_problem = cp.RadiationProblem(immersed_mesh, radiating_dof, omega)
radiation_solution = solver.solve(radiation_problem)

# Define free surface
x_range = [-radius, radius]
y_range = [-radius, radius]
nx = ny = 64
free_surface = cp.FreeSurface(x_range, y_range, nx, ny)

# Compute free surface elevation for both diffraction and radiation results
diffraction_elevation = cp.compute_free_surface_elevation(diffraction_solution, free_surface)
radiation_elevation = cp.compute_free_surface_elevation(radiation_solution, free_surface)

# Add incoming waves to the diffraction elevation
incoming_wave_amplitude = 0.1
incoming_wave_frequency = omega
diffraction_elevation += incoming_wave_amplitude * np.sin(2.0 * np.pi * incoming_wave_frequency * (np.arange(diffraction_elevation.shape[0]) / diffraction_elevation.shape[0]))

# Create animations
diffraction_animation = cp.Animation(solver, diffraction_problem, diffraction_solution, free_surface, diffraction_elevation, camera_position=[5.0, 5.0, 5.0])
radiation_animation = cp.Animation(solver, radiation_problem, radiation_solution, free_surface, radiation_elevation, camera_position=[5.0, 5.0, 5.0])

# Run animations
diffraction_animation.run()
radiation_animation.run()
```

This code sets up logging, creates a sphere mesh, solves diffraction and radiation problems, defines a free surface, computes free surface elevations, adds incoming waves to the diffraction elevation, and creates and runs animations for both diffraction and radiation results. The camera position for the animations is set to [5.0, 5.0, 5.0]. You may need to adjust the parameters according to your specific requirements.