Depends. Ligament injuries are usually graded 1-3. Grade 1 means stretching or small microscopic tears of the ligament. Usually treated without surgery. Grade 2 are partial tears and are in the grey zone, may need surgery, may not. Grade 3 are complete tears and will typically need surgery to preserve joint stability.
Yes. Many people have suffered torn ligaments and recovered without surgery. The healing and results concerning pain and function depend on what ligiments are torn and to what degree, and the general health of the individual.