Translation:I don't like mint very much.
That's the "much", it doesn't explain the "very" (or if it does then which is the Japanese word that means "much" in the sentence?). Without the "あまり" it is just "I don't like mint" with it I still don't see anything that correlates to "very" just "I don't like mint much". Not that "I don't like mint much" and "I don't like mint very much" have radically distinct meanings in English.
Yes. There is a scale on which these fall depending on whether the result is negative or positive (ex: "very much vs. not really", etc.). This is a pretty good explanation: http://www.shiro.jp.net/2012/12/yoku-daitai-takusan-sukoshi-amari.html
Just be aware that とても is a unit of degree and don't confuse it with たくさん which is a unit of amount.
とても嬉しいです。 = I am very happy.
たくさんの本があります。 = There are many books.