Would it be possible to create an alternative chain where all coins are pre-calculated from the beginning on so that no mining is possible? The system should be used to secure transactions only. I guess this question is related to the question "what happens when all 21 million Bitcoins are mined?".
As stated previously. Yes, it's possible.
However, you then have to figure out how to distribute the coins. How do you propose to do that?
In Bitcoin, mining distributes the coins, based on "work" done by the miners. In fact, since there were so few transactions at first, Bitcoin used an elegant approach and "paid the miners" a finders fee if you will for doing the work to confirm transactions and build the block chain. If you participate in helping the block chain, you can get paid for doing so. It distributes coins based on work effort.
Eventually the 21,000,000 coin limit will be reached and miners will not longer "pay themselves" for mining. At that point they will get paid for "confirming" and the earned monies will be the transaction fees on the transactions that are in the blocks. With hope, there will be a lot of transaction fees so the miners will continue to "mine"/confirm.
Without miners Bitcoin dies, because no transactions will be confirmed and placed on the block chain.