This is honestly a good question that comes up at least once on every site I've been on, so I wanted to take the time to write a rather thorough answer. I've tried to keep it as short as I can while still being thorough, so I apologize if it seems a bit brusque.
Historically, polling questions have been strongly discouraged, as they tend to prevent, rather than promote, discussion. This might seem backward, but it isn't as limiting to the types of questions I think you're aiming for as it could seem.
Let's start by imagining the following answer:
We should not do this thing.
It's not actually particularly helpful, as it doesn't provide any justification or explanation. While it might be a way to measure what people think, it's not a way to see why people think it. It consequently doesn't provide for much of a chance for people to justify and discuss responses.
In other words, questions which could be written as polls almost always better serve the community by being discussions, unless there is absolutely no more conceivable discussion to be had. Here's how I might suggest reworking the examples you've given:
"Do you feel that questions quality has improved over time?" could become "What can we do to improve question quality?" or "What should we continue to do that has significantly impacted question quality over time?"
The reason I'd prefer the rewordings is because the first question prompts the response "Yes" or "No," while the other two lead to reflection, justification, and eventually action.
"Which category of puzzles do you prefer?" could easily become "Why do you enjoy the puzzle types you like?"
While this may be primarily opinion based on meta, it also could have its insight - but that insight would come not from the category people prefer, but mostly from why they do so. "Why do you enjoy [puzzle]?" is a lot trickier of a question, and demands a more insightful answer, than "Do you enjoy [puzzle]?"
The last category, I think, is two different things. "What's your favorite question?" might be better as "Why is your favorite question your favorite?" for much the same reasons listed above, though it does seem a little broad.
Burninate requests, feature requests, new close vote reason requests, et cetera, tend to be voted on based on whether people agree with the proposals and the answers posted. In a sense, for these questions, voting is polling.
This isn't to say polling doesn't have it's place, though. There are a few primary use cases for it I can think of, though I'd only recommend using them sparingly and with consideration beforehand:
- When all of the discussion has been had, there are no new points to make, there is no new information we can reasonably gather, and a decision hasn't yet been made. In such a situation, asking a "why" question would lead to a rehashing of arguments, rather than action, so asking "pick where we go from here" is viable.
- When agreed upon as useful by the community for an express purpose. This might be where our "best puzzle of the quarter"-style questions fit, for instance.
- For tiny/lighthearted/trivial decisions (like, "hey, want a hat?").
You do make some good points at the bottom of your post, though, and it would be incomplete of me to ignore them.
- re: anonymity: I can definitely see why this might be valuable. However, it only takes one person to voice and justify an opinion; then, everyone else gets to vote anonymously as-is. If nobody has a good enough justification for their opinion, maybe it's worth considering that the opinion just isn't justifiable enough to begin with. (This is an issue with polls: it leaves room for people to agree on what we're doing but hold strong rifts on why we're doing it at all, which only leads to trouble down the line.)
- re: equality: In a lot of ways, while this might not be a popular statement, the opinions of someone who's been around for a while can be more valuable than the opinions of someone new. People who have been around a while know the system better, know our culture better, and are better prepared to provide relevant discussion points. I'm not suggesting we ignore the feedback of new users or vote based on user age, as new users' posts can definitely provide valuable input, but there's also something to be said for the experience one gains by being involved with the site over time.
- re: convenience/reliability: In a lot of ways, this is actually already what we do on feature requests. If you think something is a good suggestion, upvote; if not, downvote. However, bear in mind that the body of discussion happens in answers. When it comes down to implementing something like a new close reason or destroying a tag, the written points and counterpoints matter a whole lot more than the score on the post.
Ultimately, though, if it helps us make the site better, I'm all for it. It's just one of those things I'd suggest doing with care, and typically in consultation with the community.