There is one thing about mobile ads that I can't completely understand. Let's say I put ads in my free mobile game. What counts as a successful ad view? When it is just displayed on a user's screen or the user has to open the ad in order for it to count as viewed?
It depends. The standard as of April 6, 2016 put forth by the Media Rating Council (MRC), Interactive Advertising Bureau (IAB), and Mobile Marketing Association (MMA) in their updated Mobile Web Measurement Guidelines and Mobile In-Application Measurement Guidelines call for "reasonable assurance that the ad was rendered on the device in order to count it as a valid ad impression." (as reported by MarketWatch).
That being said, guidelines may be adopted to varying degrees by providers. Generally the fine print of legal agreements, EULA, etc dictate what will & will not count toward monetization, unless trumped by law. When in doubt, read your contract &/or hire a lawyer for legal advice.
There are many ways you can monitise adverts. Different add providers will use different conditions or a mix.
The most common is click through, i.e if the user clicks through the game developer would get a small amount.
Video ads on the other hand are an example of where they would pay just for users to view the ad.