There's a shared sentiment here that while we want to help guide users to write better questions, we don't want to create a system that becomes a discouraging wall or obstacle to a new user. As has been noted in the comments we're playing around with a few changes here and there on Stack Overflow to address some of this, including potentially creating templates on the ask page and mentoring users. But those are still in much of test phases.
For stuff we can accomplish now, there's a few things.
- The interstitial shown to new askers could be turned on, and further modified to cater specifically to the needs of TeX here. We've tried these on a few sites, the success seems highest in situations with a lot of dupes available for search (roughly half of closed questions here are marked duplicates), so this could be a handy way to target users when they're beginning to ask.
- The How To Ask page (which, yes, shares the same title as the previous but is distinctly different) can also be customized to further address the needs of TeX specifically. This page gets linked to when a question is closed as Unclear or Too Broad (which 30% of closed questions here are), so this is more of a recovery for those who have already asked astray.
- The sidebar shown on the Ask Page can also be modified, though offers less room. An example of customization can be found on sites like Mathematica.
- Customized warnings based on tags used (example from Stack Overflow) can be implemented if there's particular tags that tend to draw more work attention to them.
- The Tour page is already editable by moderators.
In experience, what tends to help a bit more than a required test or a long static tutorial (while cute, the current top answer's tutorial is... extremely meaty in length) is to have well-written and maintained advice kept, either in specific places like outlined above or on Meta, that can be succinctly and conveniently linked to askers in the moments most appropriate that they need them. Places like close reasons or tag warnings or sidebar notes can all be used for this manner of linking, to provide a more readable experience.
If there's still interest in pursuing this direction, we can look at implementing changes to the sections noted above. We'd need folks to band together and figure which parts they'd like to target, and what they'd like to see as the end result though, and submit it as a complete feature request.