I use Harvest, http://www.getharvest.com/ http://www.getharvest.com/. I think it can do all of the things you want it to, except maybe import expenses via email

I use Harvest, http://www.getharvest.com/ http://www.getharvest.com/. I think it can do all of the things you want it to, except maybe import expenses via email. It also has a public API. We're very happy with it.