Yes, I see what is happening from trying it out. I create a New File and it comes with a basic locked layer called "Background" A locked layer will not allow you to erase to transparency until it is unlocked. So far nothing unusual.
1. Same here, eraser uses background color. That's because the Background layer is locked. The background layer must be changed to a floating layer.
2. Yes, same here. What happens is that it changes the background layer into a floating layer. You see the layer name change from
"Background" to "Layer 0" in your layers panel. Then it erases to transparency.
3. Because the it's no longer a Background Layer, your basic erase tool can erase to transparency.
4. The magic erase tool does the same thing: changes Background into Layer 0.
So yes, the Eraser tool can produce transparency, but the layer must be unlocked first. The eraser tool does not unlock like the others.
I don't know what you were doing before, but so far nothing is broken in Photoshop, the Eraser tool works as expected.