I believe to "live life in a minute" means that at a time of serious jeopardy or in a life threatening situation, you live all the best or most precious moments of your life in an instant. It is in that moment, or instant, that you think of everyone you love or have ever loved and question whether or not you will ever see them again. Although I have never been in that situation, but in "The Most Dangerous Game" it is a terrifying thought to be in the position of Rainsford and hunted like an animal. I also think that it is that feeling that leads to the statement that he is still "a beast at bay."
Ah, good questions. First, where does the author say that one of the characters lived life in a minute?
My version of the story says, regarding Rainsford, that "He lived a year in a minute." This means that his senses were so heightened by the adrenaline rush of one man hunting another that it feels like that minute is full of a whole year's activities. It also refers to how time feels like it slows down at moments of crisis.
As far as still being a beast at bay, that has several meanings. Most simply, it means that the "game" didn't end when they'd originally planned: the hunt is still going on. On a deeper level, it refers to Rainsford's having found a more dangerous inner self. He's both warning General Zaroff and admitting how much this "game" has changed him.
The meanings of the quotes are given below:
"he lived life in a minute"- It means that he had treasured every minute of his precious time, maybe he was having serious trouble or having a life-threatening "life or death" scenarios, all the emotions come flushing in and he begins to remember his family members and what holds true to his heart and forever never be separated and wondered that if you would ever get a chance to meet with them as your spirit rises to heaven and beyond.
"I am a beast at bay"- It means that he is using imagery to compare himself as the ferocious beast. He is trying to say that he had regained inner strength from his inner soul and that gave him power and courage to beat all odds to hunt down General Zaroff and kill him. It shows a new personality of him as earlier on, he was very timid and cowardly but now, he had found power from his sub-conscious mind and he is fighting for his life to survive in this "game" of cat-and-mouse, where the hunter meets the hunted.