Select your pink pen stroke and Object> Expand- just click OK in the dialog
Select both the expanded Pen path and the shape you want to cut (make sure it is a filled path or outlined text and not just a path with a stroke)
Go to the Pathfinder Panel and click Minus Front
This will cut the shape into separate parts.
The eraser tool in Illustrator doesn't work like the eraser tool in Photoshop. It won't work for this. I would use the Shape Builder tool instead.
Type some text, and do Type > Create outlines
Open the Pathfinder and hit the *Unite" button.
Draw a shape around the bottom half of the text using the Pen Tool, with no fill, and a stroke you can see.