Will acupuncture help tendons and ligaments in the knee heal faster?

Yes. Yes it will. I would also advise you take some glucosamine and chondroitin, be very gentle if you exercise, and if you do exercise to do it in a pool or in the bath tub. This is all in general. Every case of cause is a little different therefore it is good that you be seen by a processional.
No definite proof. Acupuncture has its place in orthopedic medicine, but I am not aware of any quality medical study that demonstrates that acupuncture results in more rapid healing of tendons and ligaments. I typically consider referring patients for acupuncture in my practice when traditional treatment measures have not resulted in optimal results. Your physician can help you determine what treatment is best.
Probably. It is certainly possile but would likely require repeated treatments. Ligaments may not heal themselves well due to poor blood supply and remain less strong as compared to prior to the injury. This may result in looseness in the joint, chronic pain, and susceptibility to more injury. In these cases, prolotherapy or prp injections can also be useful especially when precisely guided by ultrasound.

Related Questions

Can doing acupuncture treatments help tendons and ligaments in the knee heal up faster?

Yes. Yes but with the caviate that acupuncture will not fix a lesion that requires surgical intervention. See an orthopedic surgeon or sports medicine physician to determine if surgery is required.