As far as the base goes, For the most part the break between the base of the anvil and whatever base the anvil is sitting on is enough to stop any actual reflection of force through that transition in materials. However you want a massive base (massive as in mass, not just 'large') so that the anvil itself does not move at all, and thus transfers all the force back to the hammer blow, and not into translational force moving the anvil / base.
A concrete platform is a very bad idea, as a good anvil will essentially strike it's base with an equal amount of force to what you hit the top of it with, and you will crumble / shatter the concrete in a very short amount of time.
The ring is all energy which is not being sent back into the hammer blow in the first place, so dampening it does not have any real apreciable effect upon the performance of your anvil. So you are correct, a sturdy base does help preserve the transfer of force back into the hammer blow, but not 'directly' as having a more massive anvil would, but indirectly, by helping to preserve and controll the direction of work being done. (work in the physics sense, force times distance)
EDIT: To add. It is possible with an anvil with a nice flat bottom, to get a good amount of added force rebound by placing the anvil ontop of another solid object with good rebound (ie another large block of steel) if the contact surface is regular enough. The return isnt nearly as good as just having a larger anvil, but there can be some benefit.
