Is it possible to have the separate diagnoses of carcinoid syndrome and sarcoidosis at the same time?

Yes. These are two separate diseases. Carcionoid syndrome is flushing and diarrhea associated with a carcinoid tumor and sarcoidosis is a disorder in which the body makes granulomas, typically in e lungs.
Possible. Sarcoidosis and carcinoid have been looked at together. There are patients who have had both. Sometimes sarcoid lesions have been mistaken for metastasis, which was clarified by biopsy. Tumors can cause a reaction that looks like sarcoidosis. I highly advise working with your oncologist and other specialists to sort this out.
Certainly. The two are two unrealted entities. Carcinoid syndrome (some symptoms are hot flash and diarrhea) is caused by hormones secreted by a rare type of cancer called neuroendocrine tumor. Sarcoidosis is not a cancer but rather an inflammatory condition where clumps of inflammatory cells aggregate in the lymph nodes, lungs, liver, or other organs and may or may not cause symptoms.