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I disagree here. Both look completely impossible to me.

No.

There is no any particular "rule" here. It's just that there is the word поп and there is the word попок. But it doesn't mean that words like смирнок, козлок, новикок, and others exist. They...

Why do you think Popkov sounds weird? It doesn't.


Если бы фамилия происходила от слова "попка", то ударение было бы другое - Пóпков. А скорее всего вообще было бы Пóпкин. :D