How long can people from Ukraine officially stay in Germany without a residence permit or other official papers?
People from Ukraine can enter the country without a visa and stay for at least 90 days, an extension is possible. The Federal Ministry of the Interior has compiled further information on the stay for people from Ukraine in Germany. link

Do I have to take people to the office or other administrative procedures?
No, there is no such obligation.

Who vouches for the persons admitted?
The federal government is currently developing regulations for granting protection to people from Ukraine who enter Germany.

Do I have to inform my landlord about the additional people in my apartment/house?
In any case, this is advisable.

Who is liable in the event of damage? What about insurance claims?
As in any case where someone culpably causes damage, the person is liable for it; if insurance is in place, it may materialize.

What happens if the refugee falls ill? Do you have health insurance?
All arrivals should find out what steps they need to take upon arrival, including the question of health insurance.