Yes, there is a shuttle to Golden fron Denver International Airport. You can take the light rail there too.
I'm not sure, but I would just use uber, it's only $20-$25 and 10 times faster
If you need especific information, you can ask the Police on the airport. They can help you.
There is a commuter train but I dont know its route. Try this http://www.rtd-denver.com/index.shtml
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