Smooth Voxel Planets?

So im pretty much a noob regarding voxels, but I understand the basic concept of how they work.

So my question is: Can you generate a "planet", like this using noise (obviously with actual mountains, caves, etc) and have the center of gravity be the center of the planet?:

Assuming you can, then obviously some strange things will happen as the player moves away from the 6 "center points" of the planet (like the ice cap). The player will be walking over the corners of the voxels because the player's y-axis will no longer exactly align with one of the voxels faces.

Now, second question: Will this this problem be solved by rendering using something like Marching Cubes to achieve smooth terrain?

Furthermore, how will destroying and building at run time work using Marching Cubes? My thoughts are that you might remove or place a piece of terrain next to your click because you are not clicking on the center of the voxel. I know there is a solution to this since iv seen and played games with this system, but an quick explanation would be nice.

Thank you for your time
EpEp