Is it normal for my hip to feel tight after a thigh injury?

Possibly. A couple of thigh muscle originate about the hip joint. If you strained any of them it may cause hip pain as well. If you are limping, this may cause hip pain also.
It depends. After a joint injury the joint will tend to stiffen up during the inital healing phase while its relatively immobile. This is more common in shoulders and elbows. In the hip, some stiffness is expected, but it should resolve by 2-3 months. Try stretching exercises or yoga. If it fails to improve, it should probably be evaluated by a trained professional and may need a little rehab.