What is a good way to cut hard plastic so it will have a smooth finish?
I would recommend cutting the peg(s) using a fine toothed saw like this one:


These saws cut very smooth and are made by Xacto. Should be available from most serious hardware stores and craft centers. 

One Source:

http://www.sears.com/x-acto-extra-fine-saw-blade-tools-hand/p-SPM12429862532

Many types of plastics can be sanded with very fine sand paper. You want to look for a product known as "wet or dry" sand paper with a grit number up to 1500 or more. If the cutting operation leaves some saw kerf marks on the stub of the peg you may need to start with a courser paper and then work your way up to the highest numbered grit to get the polish you desire. The Silicon Carbide type wet-dry sandpaper should be available at good hardware stores. Sometimes you can also find this at auto parts stores.