Let the court know you moved. Make sure you have a forwarding address at the post office.

This is almost certainly a scam.....................................

If it is not in writing you have a tough case. Your sister can just say that you agreed to pay half, and have not done so.

That depends on the laws of Germany.............................

Show the new lawyer your settlement papers...............

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Depends on what happened at the trial level, and whether you are still within the time frame to appeal.

The judge is not required to "ask for verification." You (or your lawyer) should have objected. If the judge wrongly denied the objection you may...

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Please see the prior answers to your duplicate question.

You could possibly retain a private investigator..............