@anton_kozlov thanks for reaching out and starting this thread. I’m glad to see you’re as interested as we are in running high quality hackathons 
Your questions are interesting, and there is no simple answer to any of them. Hackathon formats differ from field to field and organization to organization. If you can tell us more about your goals for the “Green hackathon”, we could help with a concept - something I’ve recently started doing professionally.
Here is my quick take - and I would welcome differing opinions from others on this forum:
TF 1. What if a person could not find any team members?
Most important is that everyone can easily approach organizers for help. It’s a case-to-case thing: some people need to a little extra encouragement, others need to be gently told to open their minds to other ideas and not fixate on a, let’s say, unpopular problem.
TF 2. What if there is a person in a team, who initially liked the idea, but later it turns out that the vision of team members is different and there are some conflicts in the team with that particular person. How to address this problem?
The same way you address conflict in any social or professional setting: a pinch of inspiration, a ball of gumption, a dashing smile. Get everyone to put together the pros/cons of their idea on Post-Its and look for compromise. Keep an eye on the clock - if they stay stuck for too long, suggest a split in the team. Creative chaos and safe place for ideas to conflict is what hackathons are for!
TF 3. Should there be some “advisers” (i.e. people who are not members of any particular team, but are more than happy to help brainstorm, or advise on any technical problem)?
In my experience, this can be invaluable. The advisors need to be coached though, to ensure that everyone understands the wider goals of the hackathon and to avoid certain uncomfortable situations, which often come out of mismatched expectations.
TF 4. Should we gather some ideas in advance? (like a starting point for brainstorming)
I’m all for it
TF 5. What if there are no ideas whatsoever? (extreme case, but still)
Can’t comment on something I’ve never experienced, sorry.
SE 1. Should there be any talks/presentations during the event, or is it better to let people work on their projects without distractions?
Talks help to break up the pace and a detailed scheduled helps give people an idea of what to expect. There will always be people who just want to keep working, so make sure the talks aren’t too loud/distracting - ideally in a separate room.
SE 2. How long the event should last? (best practice scenarios)
1-3 days is typical. I’d love to go on a Startup Bus once, though!
SE 3. What is the general structure of a Hackathon?
Start -> Hack -> Present -> Finish. Sleeping and eating highly recommended, but optional.
EH 1. How the decision on the “best” prototype is being made? (e.g. voting?)
Here in Switzerland we practice a form of direct democracy called voting with our feet. I.e. either your project gets used, promoted and supported - or it lies dormant in the wiki until someone comes along who “gets it”.
EH 2. What would be a good award? (e.g. a certificate, some package from sponsors – is it enough?)
Ask yourself what makes you work harder: certificate, some package, … and use it to motivate your participants. People are different in many ways, but when it comes to the fundamentals we’re quite alike.
EH 3. What to do with the prototypes afterwards: should we as organizers somehow assist in the improvement of prototypes or is it up to teams, what to do with the prototype?
Ah, the million dollar question! See http://blog.datalets.ch/000/