Let's suppose I want to collect stock prices for let's say AAPL on NASDAQ.
Where do I find information on how often an official stock price is calculated? Is it every millisecond, every second, or not that often?
Does the public have the opportunity to get every officially calculated price, or do we only get some price calculations like every minute or so?
Stocks prices are determined whenever a buyer and seller agree to trade at a given price. The company (you use AAPL as an example) doesn't set its own stock price. Rather, the investors set the price every time it trades. There's no "official" price -- just the last trade.
Likewise, you can offer to trade a stock at whatever price you want: that's the definition of a limit order. You might not find a willing buyer or seller at that price, but you can certainly open an order.
Stock quotes that you get from your broker or a finance web site reflect the price as last traded. These quotes are updated throughout the trading day and the frequency and delay varies amongst quote providers.
Like Knuckle-Dragger suggests in the comments, there are ways to get real-time quotes.
It's often more helpful to think in terms of bid/ask instead of "official price". See this question for details.
