Bill and I have a dog . What's the usual way to say this:

That's Bill and my dog. / That's Bill's and my dog. / ...

Using pronouns:

That's you and my dog. / That's your and my dog. / That's yours and my dog. / ...
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Comments  (Page 2) 
WaïtiHiya Kooyeen
(long time no see, buddy, btw)
I think I'd say :
"that's Bill's dog and mine"
and using pronouns
"that's yours and mine" or "that's your dog and mine"
However, in case I need to remind you : I'm not a native !
It may be safer for you to hear from an informed Grammar God.

"Whose dog is that"?

"It's our dog"

"Our dog"?

"Yes, it belongs to Bill and I", or more commonly "Yes, mine and Bill's"
I guess the situation I had in mind is about you showing a picture of two dogs sitting side by side, and identifying them : "that's bill's dog and mine".
I take it this is not the situation you asked about.
And even if it were, then I suspect the correct way of saying would be "that's bill's dog and that's mine" while pointing at each of them separately.
Just thought I should clarify... Emotion: wink
Site Hint: Check out our list of pronunciation videos.
The Chicago Manual of Style had a Q&A on this this month:

Q. When referring to the house belonging to my wife and me, I have trouble deciding between “Libby and my house” or “Libby’s and my house.” Which is correct?

A. “Libby’s and my house.” In some contexts, the difference could be critical. You might not want to say, for instance, “We put Libby and my house on the market.”

Source: http://www.chicagomanualofstyle.org/CMS_FAQ/new/new_questions01.html