Account-Creation Restrictions

Blacklisting one-off email services like Mailinator is as far as we should go with account creation restrictions. These services have a generally negative connotation: their users intend to sign up for something they know nothing about or even distrust, or are trying to accomplish something insidious (like ban-evasion). I'll also add that, if a new user has that kind of fire-and-forget attitude about SO, it's a good time to educate them on why the address they chose isn't allowed. Doesn't have to be long, just something to the effect of "We want you to stay around and help our community grow and cultivate the most righteous Q&A! Please use your real email address." Such sentiments will be lost on the ban-evaders and vote-gamers, but hey, users new and old should receive feedback for their actions.

Question-Asking Restrictions

Below, I outline three common question-asking restrictions and my thoughts on them. In general, I don't think the impacts of these kinds of restrictions would be positive enough to justify.

Minimum Rep

Forcing users to answer questions or edit posts before they can ask their possibly amazing question is a terrible idea. I think the gamification and vote-fraud that this solution encourages would die down over time, but only because of how many innocent users it has turned away from the site in the process. There have been other cases where a trial period has been implemented to gather data for the justification of a suggestion--I'd be curious if this suggestion has received such a trial period in the past, but I would personally hesitate to even try.

Minimum Age

We received some interesting data from Shog's answer on this front.

                Account Age | < 1 wk | >= 1wk | % of first questions which | 37.37% | 30.86% | were closed or have a negative score 

Before seeing Shog's data, I was originally against this idea. I'm still on the fence: there may be some recoil related to forcing this behavior compared to the users who did this voluntarily, and I still worry that well-researched, burning questions would be lost forever during this trial period. However, in the interest of sustainability, I think this could be one solution we at least try to gather more data for.

This approach differs from the previous rep requirement in that we're not demanding anything of the new users.

Hey, take a look around! We suggest that you take the time to learn how the site works over the next week or so before you ask your first question.

feels a lot better than

Hey, take a look around! Please read this documentation about reputation and how to get it; now go get to work! If you don't give us something useful in return for our services, we're never going to allow you to ask a question here.

Leading into my alternatives, I think it's important that, if this suggestion is implemented, the idea that "You have lots of options to find a similar question to the one you want to ask right now" needs to be stressed during this "trial period".

Minimum "Approvals"

The idea here is to allow users to post any question freely, but remove its visibility from the site until a triage-like voting queue allows it to pass through. Another interesting idea with possible drawbacks -- let's get some data! My initial reaction is to not go in this direction, as it is both misleading to users and potentially frustrating with minimal paths to feedback. To take the sustainability perspective again, creating a second triage queue actually sounds pretty silly, but I admit I may be missing the point here.

Alternatives

Incentivize dupe-finding

This section of my answer already has already been discussed in other answers to this question (and many others).

I think it's important to help curb the overflow of answers to bad questions by offering the gamifiers a more constructive path to their goal, as defined as better-aligned with the community's goals. Namely, give rep for successfully identifying a duplicate. We could even emphasize this in the how-to-answer section, explaining the ways to seek out dupes and why finding a duplicate question is better overall for the site than answering localized, no-effort questions before searching for existing resources. This may encourage unnecessary dupe-voting, but I think the trade-off leans in a very positive direction for the site in terms of sustainability.

As a side thought, users with enough rep for a dupe-hammer shouldn't need this incentive at all, so we can safely assume that questions will only be accepted as dupe under reasonable circumstances. This would assuage the gamifiers' inevitable attempts to gamify... of course, it's also wishful thinking, so something like this deserves a data-collection period as well.

Search-first UI priority

"Wait, what's a tag wiki?"

-me reading a random meta thread nearly a year after joining SO.

There are many resources available on this site that are unfinished, unfindable, and/or unexplained for the uninitiated. Sure, research can happen before a user makes it to SO, but we have to get it across early on that this isn't just a Q site.

Now I will concede that incremental learning is the natural progression of things, and that we cannot expect to make new users experts who have assimilated all opinions and culture from meta before they post their first question. However, with the poor-quality question tsunami in mind, we could at least try modifying the UI to help users find answers to their questions that already exist.

Users in general tend to follow the "path of least resistance" to accomplish what they want. I think suggestions like this one (which modify the path of least resistance from clicking ask, submitting, and waiting) are the least destructive way to encourage new users to use the resources available to them before posting a horrendously under-researched question. We don't want to restrict users from asking their bad questions--instead, we should first help them see what other options are available first.

As an aside specific to tag wikis, I would add to the above suggestion by saying that tag wikis should be a #1 priority to show new users who haven't used them yet. Not all users randomly click on everything until a magical answer bag falls into their lap--in fact, users seem to have learned that asking terrible questions gets them that magical answer bag due to SO's timeless gamification dichotomy (or, more accurately, codependence.) Anyway, I think it'd be beneficial for all to know about these easy-to-access-once-you-know-they-exist knowledge repositories from the beginning.

re: my quote above about tag wikis... Once I had read the thread that mentioned tag wikis, I went straight to the help center to find out what the heck they were. Nothing. I searched SO's meta. Nothing. I used google to search SO's meta: finally.

Conclusion

To be frank, it sucks that we have to have conversations like this one where we consider locking users out of this awesome platform. My hope is that educating new users is placed on a pedestal in lieu of restrictions and limitations.