I might be able to help on both fronts, changing the color of the edge and deleting it (make transparent).
First you're going to want to make sure the image can have transparency.
In PSP you have two types of raster layers. Raster and Background. The difference between the two is Raster can have transparency and Background cannot. When you first open an image in PSP it opens it as a background layer unless it already has transparency such as a PNG could.
The Delete button on your keyboard performs the "Clear" operation, "Clear" can also be found in the Edit menu. "Clear" deletes pixels from a layer leaving only the transparent areas. If the layer is a background layer it instead performs a "Fill" function where it takes your background color and replaces your selection with that color.
Once you have the edge selected with the Magic wand tool all you have to do is decide if you want to make it transparent or change the color.
If you want the color changed make sure the image is a "Background" layer choose the color you want on your materials palette for the background color and hit the "Delete" key or choose "Clear" from the edit menu.
If you want to erase the color make sure the layer is a "Raster" layer. You can do this by right-clicking on the background layer and selecting "Promote Background Layer" or by doing so from the Layers menu. Then simply hit the "Delete" key or choose "Clear" from the edit menu.
Pretty cool how the type of layer makes all the difference here isn't it.
