I agree with Hagar. You dont get noticed or questioned by any authorites as long as you dont annoy anybody. Added to that, if your nationality is European/Western/American, then nobody raises eyebrows!

Wearing a fake wedding ring would make no difference (ofcourse you can fool the passersby) :-), as the authorities here go only by authenticated printed documents, if at all they would need any.

So be cool! Visit and enjoy your holiday in Doha!