Can heat sensation and/or swelling in the feet be caused by diabetes?

Possibly. Diabetes can cause neuropathy, which can cause many different neurologic signs such as pain, numbness, tingling, or heat/cold sensations. Swelling is less likely directly from diabetes, though diabetics often have vascular problems that can contribute to it.
Many possibilities. Symptoms of neuropathy can range anywhere from numbness, tingling, or burning sensation, to an uncomfortable painful sensation of the extremities. For some, these symptoms are mild, while for others, they can be severe and disabling. Neuropathy can develop as a result of a pinched nerve, trauma, drug side effect, and some diseases such as diabetes. Get evaluated and professionally treated.