Can rehab allow someone to live independently?

Yes. Depending on the reason why the person needed rehab in the first place. The goal of any rehabilitation is to improve someone's function to the fullest it possibly can be.
Yes. I would say that it has to be successful rehab. You just can't do rehab, you have to do it and progress and improve, and depending on your impairments you might need some other adaptions. The person in a wheel chair needs different things than the person who can get around with a walker.
Varies. Patients can have variable responses to rehab/ physical therapy. It is partly based on the injury or condition itself, but also based on the patient themselves. For example, the rehab potential for a knee injury in someone who is otherwise healthy and active is different than that of someone with the same injury but is elderly, has dementia , and does not follow instructions.