If the root is dead, and the crown can be lengthened, can the tooth hold a new crown? Attached is my x-ray (Photo)

Your dentist is in the best position to determine if a crown lengthening procedure would allow long term success with a crown. Otherwise, you can have a nice crown that would fail shortly.  X-ray will not show that. Besides, x-ray will not show if a short implant can be placed without sinus augmentation. If you have some good bone, a crestal sinus lift can be done through the socket to add some bone height. Crestal sinus lift is not really an invasive procedure. 

There are basically 2 options. One is to do a root canal / crown lengthening and a new crown. Based on the x-ray, it looks like this option does not have very good prognosis. It seems like too much of the tooth structure has chipped , too close to the bone. Your second option with better prognosis is the implant ( given you are a good candidate for it).

Your tooth is definitely infected and if it can be saved will need a root canal.   The X-ray provided is a "periapical" and is not the type of X-ray to determine the location of decay or fracture in relation to the bone.  This is due to the angular on of X-ray; ideal diagnosis would require a bitewing X-ray as well which gives a more true bone level location.  If you would like a second opinion I would suggest a referral to a periodontist for evaluation.  Periodontists are bone and gum specialists who are able to perform crown lengthening if it is even required. I believe your dentist is correct in regards to the sinus location which would most likely require a sinus lift in order for implant placement.  If this is needed, the periodontist can discuss and perform this as well.  If indeed the tooth cannot be restored and you do not want to go through the implant and sinus lift procedure you could restore the area with a bridge as well.