Would sports also increase the rate of division? Bodybuilding for example you intentionally make micro tears in muscles to get them to repair and grow (lay person, apologies for the mistakes)
Hmm, I'm a molecular biologist, not an expert at this particularly, what I do know is that damage of muscle fiber may not be necessarily mean new cell growth and thus extra DNA replication. The growth of the muscle cell certainly does.
Exercise is good for the immune system, this helps clear cancer, helps stay away from inflammation (ie auto-immune diseases). Where as a sedentary life style a poor diet will promote this.
But indeed for some things/behaviors, the relation the cancer may be complex.
The problem is that many things that reduce the probability of cancer increase the probability of other diseases.
Therefore there is no good solution for this.
For instance, some studies have demonstrated that certain kinds of malnutrition, i.e. the consumption of inadequate amounts of protein and/or with bad amino-acid profile, inhibited the development of tumors caused by carcinogen agents (in rats and mice). This was not really surprising, because tumors grow faster than the rest of the body so they need to synthesize more proteins. If they are starved, they grow slowly and the immune system has time to react and to eliminate the incipient tumors early enough.
Despite this, starving yourself in the hope to avoid cancer would be a very bad idea. While the probability of cancer is lowered, the probabilities of low resistance to infectious diseases or of muscular weakness leading to falling and breaking some limb, or of cardio-vascular diseases, become higher.