Yes, both sentences sound natural. Explanation: "by process of elimination" The phrase is used without an article as a fixed expression. It emphasizes the method used. "the process of elimination" The definite article "the" is used to refer to a specific, well-known method (the process of elimination). Both forms are correct, just slightly different in tone.

"A process of elimination" works better in both sentences.

yes! these sentences sound very natural and are most likely used often.