May I stream, record and upload gameplay of 12 orbits, or use it at public events, without any restrictions whatsoever?

Sure thing!

Can I request keys for that purpose?

Indeed! You can request keys using keymailer, woovit or distribute(). You can also contact me via games@romanuhlig.com. Please use a verifiable email address to do so, for example one that can be found on your website or YouTube account. This is the internet, so stating your name unfortunately won't do.

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.