I've found a solution!
*does happy dance, to the delight of the dog and the disgust of the cats who were peacefully sleeping*Put into your family inventory any 'personal' items that you wish to keep. Move out (selling furniture) into an empty plot.
Immediately move back, choosing furnished house.
Your house and its contents will be as you left them* and the mailbox will be working. It may cost a little because of the depreciation factor but otherwise everything is the same.
Chester was
hit with this problem following the recent meteorite strike against the side of his house. (
http://www.carls-sims-3-forum.com/index.php/topic,3333.0.html ) I didn't realise there was a problem until a couple of days after sending off several jewels to be cut, including a pink diamond. He's just moved back in, I've checked the mailbox and, delight of delights, Chester has just collected a heart-shaped pink diamond and a complimentary copy of his latest novel.
Note, I checked the mailbox on the empty lot that he moved to - there was nothing in it; it wasn't until he moved back to his original house that the items appeared in the mailbox.
The trashcan/bin also appears to be working properly. At least, I was able to get Chester to recycle the newspaper into it.
* Well almost as you left them - other bonuses included a clean sink and counter and the burnt window was as new - but I expect that he 'paid' for that. Apologies because I didn't think to see just how much it cost to move out and back.