Cut bit by bit to get the idea of how much you'd like to trim. Woven back fabrics can be trimmed as much as you like since the backing is solid. Consider a five o'clock shadow or stubble effect. For most collectible teddy bears, glass eyes are used. If the bear is intended for a child, then use safety eyes.
Most eyes are hand made and there is a wide variation so match the eyes at the time of purchase.
If you will be making many bears and are quite a distance from your supplier, then purchase a dozen pairs and match 'em up. Form a loop and then twist 3 or 4 times. Do other eye and then separate with wire cutters. Positioning eyes takes time. One easy way to use eyes on a wire with no loop.