What is this hard, rubber-y lump under my skin? Is this a sign of a tumor?

Impossible to say. Without actually examining it and possibly getting a biopsy, it's impossible to say what you have. Based on your description of a rubbery lump under your skin, my guess would be either lipoma or some cyst like epidermal inclusion cyst. Those are benign. But you may have a rare soft tissue sarcoma, which can be dangerous. You should have it checked.
Maybe not. There are many causes of "hard, rubber-y lumpy dumpy lump" under the skin. It depends on location and appearance. It could be a lymph node, lipoma (fat collection), hematoma, foreign body, neuroma, scrofula, or other causes. See your physician. They will run tests and may obtain a biopsy to determine what it is. If it is near a tendon, fast growing, and impairs function, surgery is indicated.
Probably not. Cancer would not be the first thing that i would think of. Most likely this is a cyst or a benign fatty tumor called a lipoma. You should however bring it to the attention of your friendly primary care physician.