I you decide yourself to quit a job you would normally have to wait 3 weeks before you can begin to get money from the A-kasse.
But with you it is maybe special situation because you have illness in yours family. Sometimes A-kasse can decide that you should not wait for money if you have this special reason to quit job. But please ask your A-kasse about that.
And if you move to another country you can only take benfits with you for 3 months and only for seeking job in another country.
Hope it helps you.