You are absolutely right, and that is my mistake. The game either did not tell me, or I did not see any explanation about needing to right click while using the hammer. This has opened a whole new world to me, and I'm logging more hours as we speak, and will update my OP shortly.
However I do still see problems using the axe to chop the tree, or attacking the shark with the spear. The spear does work, but only if you have saved your game and re-opened it. If you play for too long, the spear will glitch and become unusable