Will a chin implant make my face longer?

It could. Most chin implants are placed to make the chin protrude more, not make the face longer. There are many different shapes and models of chin implants and your surgeon should select the one that will accomplish what you and s/he want. If the chin implant extends below the edge of the jawbone, it could make the face look longer.
No. A chin implant should not extend below the lower border of the jaw. It depends on what you are trying to accomplish. Do you want your face to be longer? If so, you should have a sliding genioplasty with vertical augmentation and a bone graft, which will predictably lengthen the face. If you want your face shorter, the chin can be vertically reduced. It all depends on your particular facial shape.