Can a decayes tooth be infected and cause swelling with pain? can an untreated infected tooth cause heart infection?

Yes: Yes to both. Decaying teeth can cause swelling and pain and may even lead to jaw bone infections. Also, the constant source of infection from an untreated infected tooth is a well known cause of endocarditis, which is the heart infection you are talking about.

Yes and yes: Whe decay reaches the nerve, the nerve gets infected and the infection will spread past the root into your skull. Once inside your body the infection can spread into the face and eventually into your body. Tooth infections can kill if left untreated.

Decayed tooth: Decayed tooth can most certainly be infected. Can cause severe jaw infection. Can predispose to severe pneumonia including empyema. Cardiac infection possible usually requires abnormal heart valve as predisposing factor. Requires definitive treatment antibiotics and possible extraction.