If I am level 6 with the attack progression +6/+1, can I use the same weapon to attack twice during a full-round action? If I am level 11 and get +11/+6/+1, can I use the same weapon to attack three times?
Yes, you can
The only time the game cares about which weapon you’re using is if you are using Two-Weapon Fighting (the bonus attack must be made with a different weapon from the main attack) and natural weapons (Claws, Bites, etc.).