The main question boils down to whether suicide is unethical, given that free solo climbers have a significant chance to die while climbing. If suicide is unethical, putting one's life intentionally at risk should be unethical as well, paralleling that putting somebody else's life at risk is unethical because murder is.
Whether suicide is ethical is, of course, debated: There is a tension between the unanimously acknowledged value of life and one's right to self-determination. Cultures who assign a high value to individual freedom, like the modern Western societies, typically don't find suicide unethical.
The question about an indirect "advertising" effect is harder to answer because there is potential damage to others. Popular real or fictional characters certainly invite people to copy their behavior, including suicide. In recent years media coverage about suicides has become more careful and is accompanied by counseling contact information to mitigate any effect of unintentionally triggering more suicides.
Again the answer depends on what importance we assign to the freedom of individuals, both the climber's and the potential copycats'. Western societies with their focus on individual responsibility are more lenient than more traditional societies: Individuals have a lot of leeway how they go about their life, even if it may have negative side-effects. The potential copycats, assuming they are capable individuals, are equally responsible for their own actions; we are not responsible for others unless we abuse inabilities or dependence or use coercion. I also take issue with your labeling free soloing as "bad behavior": That judgement is dependent on the answer and should not be in the question.
The question whether it is unethical to endanger rescue personnel is similar to the copycat question. I can imagine that non-solo climbing also causes a significant number of rescue operations, and perhaps more urgent and hence dangerous ones because there are still lives to save. Do we find roped climbing unethical as well?
Another argument is that rescuers are volunteers; this is a different situation than, say, flying aircraft stunts over populated areas where potential victims don't have a choice.
My take as a member of a Western society which puts a lot of value on the individual freedom to lead one's life as one pleases is: Let them climb. It's fun and they hurt only themselves.