How do we know how many BTC have not been spent since being mined? Any tool to see this information? I would like to know how many BTC have not moved from the block they were mined in - never transferred or spent. (With this information I believe could have an idea of how many BTC have been lost). Is there a tool to see this kind of information?
I don't think your question quite makes sense. Bitcoins aren't "transmitted", they simply sit in the block chain until someone chooses to spend them. There's no way to tell if a particular Bitcoin is lost or not because that depends on whether anyone knows they key needed to claim it. That key could be written on a piece of paper for all we know.

When your client tells you that you have 50 Bitcoins, it is doing that in a purely passive way. Nobody can tell that your wallet contains the keys needed to claim those Bitcoins, and this is by intentional design.

You could fairly easily measure how many coins hadn't been spent since they were mined. But that wouldn't really tell you anything. Some of those coins may have lost keys and some may not. And some coins may have been spent once and then lost.