If there's some gouging evident on the surface left behind, don't worry. You can use a wood filler from Home Depot that can then be sanded down with the rest of the area to make it completely smooth.
You can get Liquid Nails from Amazon for $4, DAP paintable latex caulk from Lowe's for $3, and Bondo wood filler from Amazon for $27. To add height to the door, you'll need two or more ...