"I couldn't even process what the teacher was saying."
can I change "process" to "register" there?


plus,
"I registered the shadow as a young man."
can I change "register" to "process" there?

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Registering is usually when you collect an information into your mind. It could be used, but it would seem sort of formal or as a mock of formality. Processing is how you would understand something so...

"I couldn't even process what the teacher was saying"
Process or register could be used here but process is more common since it is simpler and since you are trying to understand not gain something but both are correct.

"I registered the shadow as a young man"
Both could be used here but since you gained information registered would make more sense but both could be used here. I dont really know how to explain it but in both of these circumstances both could be used. Once you are more used to speaking,reading hearing, you will know when which words are better than others.