A good bushcraft knife is better than an axe for survival purposes due to its superior versatility. However, this choice will mostly be decided by personal preference.
An axe will most definitely cut through bone, and different axes will perform this task better and more effectively than others. The cut will however not be clean and therefore an axe would not be suitable in all situations for this task.
Burnt wood or charcoal has the potential to hold heat. This means that if you can apply a heat source to it, the charcoal can eventually become red hot and will have sufficient heat to ignite a fire. This has application for emergency situations or where you have issues with your tinder!
When considering whether the BIC or Zippo lighter is best for survival, you need to consider the survival conditions you are most likely to encounter and pack the lighter that best fits your circumstance. Or, if you are a true survivalist, pack one of each in your gear!