The first sentence is a question (interrogative). The second sentence is not a question, and should not have a question mark. A negative question would be "Did we not play?" or "Didn't we play?"
Knowing this, and knowing that we invert the subject and verb (or auxiliary) to form a question, a positive indicative sentence "We did play" becomes "Did we play?" as a positive interrogative sentence.
PS: Centering some of your text like this will make it very hard for some users to read. The usual way to write you example sentence is to put quote marks around them.