Does sprained tooth syndrome or peridontal ligament strain need a root canal?

Most likely no: While your tooth may be sore, it will more that likely feel normal after a week or so. That injured tooth is more at risk for a root canal sometime in the future, as the trauma from the strain can interrupt the blood supply to the tooth and cause a cascading inflammatory reaction in the tooth's pulp (nerve). The older you are, the more likely it will be irreversible and require a future Endo.

Not always: Give it some time to heal by eating a soft diet and trying to avoid stressing that tooth even further. Time will tell if you need endodontic therapy.

No: The ligament that holds the tooth into the bone and the nerve/blood vessel bundle that enters the inside of the tooth are separate and distinct tissues. Trauma can cause damage today that doesn't become evident for months or even years. See your Dentist to establish a post-traumatic incident baseline.