Is it okay to add old, stale bread to the compost pile? What about tree stumps – can they be buried in the pile to speed their decay?

No, bread should not be added to a compost pile. As far as tree stumps, I wouldn’t add them to the compost pile. They will eventually decay in a long, long time, but in the meantime you will upset the balance in your pile and you won’t be able to turn it to speed the decomposing process. A balanced pile needs both green and brown materials. I would chop it up and use it for a nice bonfire, or maybe rent or borrow a chipper and use the chips around your garden, with maybe a small portion of the chips added to the pile.