Will there be an "initial price of a chain fork" type of opening when this new BCC comes out? Or the price will be same to the regular BTC that we currently use?
The initial price of BCC will be what the first buyer is willing to pay for it on an exchange. After that, it's simply a matter of demand and supply.
According to coinmarketcap.com, BCC futures are currently trading for $332. This is probably the best indication of what to expect once the market opens.