Scantron Is Not An Option

You can do this with some fairly straightforward add-ons to google spreadsheets. copyDown will automatically run a formula (or a few) to determine someone's score, and formMule will automatically send you an email with the results at a predetermined time!

What about http://quizlet.com/ I don't think you can centrally manage the grading, but it would probably be easy enough to have competitors bring their smartphones to the front to register their grades.

Another option: jotform.com has a http://www.jotform.com/form-templates/quiz-form-with-a-calculated-number-of-correct-answers, or you can make one from scratch. You can set it up so that you get an email notifier of name & score, and/or sync the results with a google spreadsheet. The are only free for up to 100 submissions per month, as a caveat if your trivia night is really popular.

I've already discovered Flubaroo, which is half the battle just in quiz-grading alone. However, considering that it's a 50/50 shot whether Google Sheets will actually cooperate on my phone, adding Flubaroo to the mix in real time seems risky. What I'd really love is something that would allow Flubaroo to run itself and then ping me the results. (Whether via email, text, push notification, whatever.) Having everyone bring up their phone might work OK for now, while we're a very small operation and typically get 3-5 audience members playing along. But the goal is for more than 10 people to come to our shows. And getting 10-20 people onstage to compare phones isn't feasible. Jotform, on the other hand, might be an option. Especially if it's as user-friendly for the general public as a Google Form is (especially via smartphone), and if there's a built-in mechanism to have everyone's score emailed to me.

Qualtrics should be able to do whatever you want in this regard. It's very extensible and the customer support has been great (caveat: I have an account through my university, but anyone can also create a free account. Not sure of the differences or if any of them would be relevant to your situation.)

https://developers.google.com/apps-script/ is basically made for this. You can execute arbitrary Javascript functions in response to Google Forms submissions (among other things), which could include sending mail from Gmail or computing and adding a value to a column on the spreadsheet backing the form, or basically anything. If you need any pointers please feel free to PM me, I worked with Apps Script a bunch a year ago.

Seconding Qualtrics.

So it turns out the best solution right now is the Google Sheets app for my phone. Using the app, i'm able to open the quiz results spreadsheet quickly and manipulate it easily so that I can see everyone's answers. I haven't figured out if it's possible to use Flubaroo with the app, but since we're averaging 5-6 participants in the "audience play along" quiz right now, manually grading the quizzes is fine. All of these suggestions are great for the future, when I can't visually scan across a spreadsheet to see who has the most right answers because we have a full house.