I don't think so on the testicular cancer.
There are two major environmental causes for cancer. The first is exposure to ionizing radiation. So do not put radiactive material into your bike shorts. The second one is exposure to chemicals, also called carcinogens, that produce free radicals. Tobacco smoke is loaded with carcinogens which is why smoking is not a good long term health decision. Well, there is another major cancer risk. That is having the wrong genes. It is thought that some people have a genetic predisposition to cancer.
The thing that ionizing radiation and free radical production have in common is that both produce mutations in one's DNA during cell replication. If the wrong gene gets mutated, that can cause the affected cell to become a cancer cell.
Notice that environmental effects like wearing hot, sweaty shorts are not one of the above. Now the accepted way to determine a cancer risk is to expose laboratory rates to the suspected problem but I have never heard of anyone putting hot, sweaty underwear on lab rats. Maybe there is a grant there.