There's absolutely no difference in meaning whatsoever, so it's just an issue of collocation (what sounds more natural.) To me, "profound" sounds much more natural in the third one. In the first one, "deep" sounds a little more natural, but "profound" wouldn't be terrible. I don't think either one sounds very natural in the second one. It would sound much better to say "a penetrating look."

Deep is broader. A deep sea, a deep colour, a deep thought Profound only works with thought or another abstract idea. Deep is physical or abstract

"Profound" and "deep" are often used interchangeably, but there are subtle differences in their usage and nuances: He had a profound/deep sense of loss. In this context, both "profound" and "deep" can be used to describe the intense and emotionally significant nature of the sense of loss. However, "profound" might imply a deeper intellectual or philosophical understanding of the loss, while "deep" may emphasize the emotional depth. Peter gave him a long profound/deep look. Here, "profound" and "deep" can both be used to describe a thoughtful, meaningful, or intense gaze from Peter. They are nearly interchangeable in this context. This is a book full of profound/deep, original and challenging insights. Both "profound" and "deep" can be used here to convey the depth and significance of the insights in the book. Again, they are quite interchangeable in this context. In summary, "profound" often carries a connotation of depth in a more intellectual or philosophical sense, while "deep" is generally used to describe emotional or literal depth. However, the distinction between the two can be quite subtle, and in many cases, they can be used interchangeably depending on the nuance you want to convey.

PS: Does either of both words sound natural?

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