The general consensus appears to be that stepwise regression should only be the choice of last resort for trying to determine a causal model.
Stepwise regression is suspect as a method for choosing a scientific model.
If the variable selection has a good scientific basis, then the final model gains credibility. Stepwise is not considered a good method, but really only a method of last resort. For COD1, COD2, and COD4, you have fewer than ten deaths each. You have too few deaths in these three causes of death to cope with this much larger number of explanatory variables. For COD3 and COD5, you may have enough deaths to do analyses on more than one explanatory variable.