Learning from the credit card fraud prevention industry
I believe a possible solution (or improvement) to this situation should actually by learning from other service, sites that are facing similar problems.
Online credit card fraud prevention is an industry that is tasked with doing the exact same thing for e-commerce sites. Their goal is to find the users creating junk accounts with stolen credit cards and making a one time purchase on a specific site - sounds very similar to the situation being described here.
Credit card fraud prevention
Most solutions rely on two approaches: Databases of bad accounts/stolen credits cards and risk scoring.
In this case, I believe risk scoring can be used rather effectively to root out most bad questions, so I'll focus on that.
For credit cards, risk scoring will look at a number of factors and assign a value to each factor, and then combine them to give a specific transaction a risk score. If the total score is above a certain threshold, the transaction will be flagged or blocked.
The Stack Overflow Risk Scoring
Implementing a similar solution for Stack Overflow should not be too complicated. Some of the variables I can think of are:
Email domain, as some have mentioned some domains are at a higher risk than others. They should just get a higher score. Lets say +5 for gmail, but +15 for aol.com (etc.)
Length of question, very short questions, or very long questions can indicate an increase in risk.
Number of tags. If there's a connection between number of tags and bad questions, it can be added to the mix.
Specific tags - is the risk higher for specific tags?
Spell checking - very high percentage of spelling errors should increase risk score.
Visits on the site before asking a question. Did the user just type stackoverflow.com and post a question? Or did he visit the site a few times today?
How many searches did they do before asking a question?
How many questions and answers they looked at before asking a question.
I'm sure people with more knowledge of Stack Overflow can probably come up with even better factors that will make it much more effective, but what's important to note is that usually a single variable will not be enough to get the transaction above the risk threshold, lowering the risk of false-positives.
OK, so we flagged a question, what now?
There are two usual ways to deal with flagged transactions on E-commerce sites. One is the completely block them and one is to flag them for human review.
This site is perfect for the second option. If a question is above the risk threshold it will just need to be reviewed before it will be available to the public.
Summary
I believe this approach can significantly reduce the number of bad questions by new users, while limiting the impact on legitimate new users.
I'm truly humbled by the level of discussion here, but wanted to add my two cents.