May I use 12 orbits in public institutions, for example in schools and libraries?

Sure thing!

Can I request copies of 12 orbits for that purpose?

Indeed! Just send me an email to games@romanuhlig.com. Please use a verifiable email address to do so, for example one that can be found on your institution's website.

What is the best game mode to get started?

There is some discussion of this in the presskit.
In short: If you are not sure where to start, try "Arena" for 2-3 players, or "Blizzard" for 4 or more. Arena requires the least amount of precision to get going, while Blizzard allows you to focus on your team and to play defensively. Of course, the best choice ultimately depends on your playstyle (or that of your visitors), but these are the easier game modes for beginners.