What are the main headaches to watch out for in an international home exchange, in your experience?

Look if nothing is broken in the home you get you do not want a claim do you? Get an inventory of the things so they won't claim you stole anything and vice verse. Be sure you have (and leave) instructions how to use the garage, use all electrical appliances, mail including instructions bills if arrive, phone calls how do you charge or pay for them. Any instructions about the garbage, many countries have different rules. I think I would go through my home before leaving it and see all the things that are done or do not want anyone to do them while I am gone. I am sure I forgot things, but make a list, have it in front of your eyes for a few days and add things as needed. Ah, tiles, linen etc. be sure there are plenty.