I would cut the stem and treat it as a cut flower from now on. I will last pretty well if you angle cut the stem cleanly with a very sharp knife and add a pinch of sugar to the water.

The plant is hardly injured at all. Give it an east window and read the FAQ on moth orchids (Phaleanopsis) and it might well bloom again in the fall.

I see... so the "Plant" is the bottom leaf part with the bulb in the ground and the stem is just the flowers' foot that grows every year?

Please forgive my awful ignorance as this is my first orchid ever. I've never had enough guts to get one before.

Therefore, do I need to cut the stem every year? And if so, at what time do I need to cut it?

Where you see the gauze on the photo is where the stem was chewed through about 75% off. Today, the flowers seem no different and seem unaffected by the damage to the lower stem but I'm still keeping it wrapped with aloe gel soaked gauze to see how she fares...

You can leave the stem (spike) on until it turns brown to try for a weaker re-bloom or you can cut it off after the flowers are gone. Cutting it about an inch from the plant will be the best for the long term health of the plant.