Is the only way for wolves to die for them to reach 15 years of age and then randomly die a week after that, or are there other ways they could die?

Nope, wolves can only die from old age (If you don't retire them). Also, the death script runs every Monday, if they are 15 then every week they have a higher chance of dying.

Wolves, as Flower Field said, die after age 15 or older in Death Script. Death Script is also known as DS. The Death Script runs 6:45 AM Game Time every Monday. The chance of your wolf dying during Death Script increases every time, however, if you would like to lower this chance you may use a Raven Feather.

Yes. Wolves die if you retire them. To retire, go to your wolf's page, click Manage Your Wolf, then click Retire Wolf. Your wolf will go to the heavens, and you will get a bone fragment.