Feeling of food getting stuck in my throat. Can the ENT check for cancer with fiberoptic cameras?

Yes. An ENT specialist should be able to examine throat and related areas. You may wish to consult a GI specialist as an alternataive.
Yes., also GI. You may have some esophageal obstruction, like schatzki ring, or plummer vinson syndrome, or cancer. A double contrast barium swallow can be done, as well as a flexible esophagogastroscopy to see what it is. Ent can check for throat cancer.