Not at first. Most breast cancers are now detected with mammograms or other imaging rather than physical examination; they are too small to be able to feel. When cancers grow, they generally become hard lumps, but these may still be moveable. When the cancer is more advanced, it can become fixed to the chest muscles and become immobile.
Yes and no. . Late cancers are fixed and difficult to move. And most cancers are hard. But some early cancers may still be moveable. Your are young so it would be REALLY uncommon for you to have breast cancer. Most lumps in young breasts are benign ( not cancer) See your MD for exam and Ultrasound. If not scary, could be closely followed without surgery. If concerning a biopsy will prove it's benign. .
Breast masses. Breast cancer masses can be both moveable and fixated. Fixated masses are more advanced (ie attached to the chest wall). Most cancerous masses (but can definitely be variable) are painless, in the upper outer quadrant, and palpable. See your physician for an exam. He/she may want diagnostic studies.