Is there AOG liability?
how long can this airplane stay in that airport?


The aircraft is "AOG" so it cannot leave. It needs to stay as long as it takes to complete the necessary repair.


  what is the airports role in this situation? Do they help them out or just charge the amount they stay on the ground?


The airport usually would charge the airline the parking fee. The parking fee is a small cost of not having an aircraft available in revenue service.