Can diabetes make COPD worse?

Yes. In addition to increases in infection risks, diabetes may be part of a "metabolic syndrome" which can include obesity, hypertension, and cholesterol problems. Obesity can certainly effect breathing. The complications of hypertension/cholesterol (heart disease) can only complicate the copd.
Yes. Havingdiabetes makes you more at ris to get infections, including lung infections which can make COPD worse.
Intriguing thought. I have seen research that diabetes can cause lung damage, and thus theoretically can make COPD worse. I don't think this has been seen clinically. That being said, diabetes has a number of complications which can complicate the health of patients with copd. These complications include heart and vascular disease, kidney disease, etc.