Yes, dogs do throw up and then will eat it again. The only thing we can do is try to pick it up as quickly as possible before they go to sniff it and then...well you get the picture and why do they do it? Maybe it still smells good, I don't know. Some other posters here may know.
Yup, all mine have done this at one time or other.
If someone loses his/her dinner we never worry about cleaning it up. We know one of them (the dogs) will be more than happy to do that for us.
I always consider it a 'save' on dog food - 2 dogs fed for the cost of one. lol