Rob, the timer you see in quiz is specific to the response that the user is responding to. In your case the the question of the day should have a close time limit which is applicable for all users. So the poll closing date is the correct option for that. Community Polls allows you to close the poll for accepting answers after specified date, quiz will not do.
However, displaying the timer is not available in polls which is available only in quiz. Also if you would like to display the link to the question of the day page on top of profile page, jomsocial should support it. Check if JomSocial has any custom module position in that place so that you can publish the link there.
For quiz, the intro page is mandatory, but can be overridden with some hacks. I can help with that if you want to go with only quiz component.