A: The answer varies depending on the type of technology, but generally between 22 and 30 months. However, there is a great deal of variation, even within each area, based on whether there are enough knowledgeable examiners on staff and how "hot" the technology is, among many other variables.
The US Patent and Trademark Office keeps detailed statistics at their Data Visualization Center: https://www.uspto.gov/dashboards/patents/main.dashxml.
The time between filing an application and concluding the application process is known as "traditional total pendency," and is graphed here: https://www.uspto.gov/corda/dashboards/patents/kpis/kpiTCTotalPendency1.kpixml.
A: The short answer is: yes, it's possible. Almost all patents are based on improving technology originally developed by someone else.
A patent can be filed on any new and useful process or improvement to a process. Playing Minecraft is a process (believe me, I know). You probably wouldn't file for a patent on actually playing the game, because enforcing that patent would mean suing potential customers. But looking at the situation in reverse, a computer performs certain processes to "present" a game of Minecraft to a player, and you can file for a patent claiming an improvement to those "presenting" processes. That patent would be aimed at a competitor running your game, not at customers (or at Mojang / Microsoft Studios, for that matter).
That being said, you can only obtain a patent on an improvement that is not "obvious" (that's that the legal term). Whether or not your particular game type improves the underlying technology in a non-obvious way is something you would want to explore with a patent attorney that you hire. You may also want to consider getting a copyright on any expressive in-game content.
A: Generally, no patent may be obtained for an invention that is available to the public before an application is filed with the patent office. The US has some limited exceptions, including disclosures made by the inventor less than one year before the application is filed, but the patent office will award the patent to the first person to apply for it (not the first to invent it).
As to your particular situation, you may benefit from tailored legal advice, but for that you should hire a patent attorney.