Will a 2-3 inch cooked carrot show up on a normal chest x-ray? What about a barium swallow?

A carrot will. probably not show up on a regular chest xray. If it is stuck in the esophagus, it will show up on a barium swallow.
Depends on place. Carrot may not show on xray in esophagus. Barium ingestion would show carrot as lucent defect in esophagus. If surrounded by air in stomach the carrot would appear dense.
XRay detection. Usually, products are not easily detected by simple xray unless they contain elements that are radio opaque. Carrots may be seen as an undefinable mass, but the actual detection would be difficult. The barium swallow may detect the carrot if it is causing an obstruction. Also, the process of digestion will have started the breakdown of the carrot.
Huh? I can't help but wonder what you're doing. Food that is chewed and swallowed, if it is visible at all on x ray, would have a very amorphous appearance as it floats in liquid stomach contents. If swallowed whole, my information is much more limited. No one would conduct a study that involves swallowing large whole chunks of food. Ba swallow would show an obstruction if it's stuck in the esophagus.