Can I play basketball with a groin injury?

Probably Not. A groin injury is usually the result of a strained adductor muscle. This muscle is used in sprints, jumps, cuts, and turns. A mild strain may not limit your ability to play basketball, but playing may increase the injury. More severe strains will limit your ability to run and jump. Treatment/activity level will vary with the of damage. Groin injuries should be evaluated for hernias.
Groin injury. All depends on level of pain and degree of injury. Often time a sports hernia is undiagnosed for months and only becomes more apparent when the pain worsens with activity. An acute adductor strain with type iii tearing will take 6-8 weeks to heal.