Do fat men get breast cancer?

Yes. Obesity is linked to a higher risk of breast cancer. Fat cells can actually make estrogen, even in men, and increased estrogen levels are associated with breast cancer. So although breast cancer is quite rare in men, obese men are more likely to develop breast cancer, just like obese women. And exercise is associated with a decreased risk of breast cancer... So get moving!
Yes. Any man, fat or thin, can get breast cancer, although the lifetime risk is only 1:1000. High-fat diets have been linked to an increased risk of breast cancer, however I am unaware of any study that has shown any increase breast cancer risk in obese men as compared to other men.
Yes. Any man can have breast cancer but, like other colleagues said, is more frequent in obese men. The rate is 1 men with breast cancer for every 100 women with it.