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Not really. If "snow" designates grass, and "white" designates green, then "'snow is white' is true if and only if snow is white" is still correct. Same if "snow" designates the sky and "white" designates green.
I'm afraid I don't understand your point.
It should have read: "Same if "snow" designates the sky and "white" designates blue."
It was apparently a nitpick of your first paragraph, ignoring your second paragraph.
That can't be right. If he both misinterpreted 'prior assumption' and made a serious typo, his comment would not have been twice upvoted, would it?
If you can't take as a given that statements actually are in the language they appear to be in no statement can have any knowable truth value. If "snow" in the utterance is a word in the same language as the identically spelled word in the statement, and the same for "white" (and "is"), and the rest of the statement means exactly the same as it does in English then the statement is still correct. But if "white" might designate orange "true" might just as well designate bubblegum or "only" designate "to treat like a second cousin".
And further the grammar of English is being assumed... as well as the very concept of languages.