Your post touches on several questions - probably the most important is the question who can edit help center and how can regular users contribute.
Who can edit help center?
As a rule, most parts of the help center can be edited only by Stack Exchange staff. Some stuff can be edited by site moderators - a more detailed description can be found here: What parts of the help center can site moderators edit?
Based on anecdotal evidence it seems that sometimes Stack Exchange makes more pages mod-editable than the default settings (on moderators' request).
It seems to me very unlikely that regular users would be allowed to edit help center. (And it is probably quite a good thing, because those pages are - in a sense - official and quite a lot thought should go into decision that something should be changed there.)
What should regular users do?
- If a regular (non-diamond) user wants to help, there is still plenty of stuff they can do. For example, many of the posts on meta tagged as faq or faq-proposed are community wiki, mainly to encourage editing by users.1 You can also edit tag-infos - both on main and on meta, but tag-infos on meta are quite a good place where to collect relevant information for some topic related to usage of the site.2 (For example, if you can see some useful additions to the
tag-info for badges tag, you can edit the tag-info.)
- If a regular user wants to suggest some change to help center, the right approach is to bring this up on meta. For things specific to MathOverflow, the suitable place is this meta. But for stuff that is relevant for all sites in the network, Meta Stack Exchange is the right place. (In the examples below, you can see some posts of this type.)
You also mentioned that you were interested specifically in badges.
For several badges you can quite easily see your progress towards particular badge, the easiest way is probably using badge tracker in your profile (although not all badges are displayed there). For more details see: https://meta.stackexchange.com/tags/badge-progress/info
In the network-wide FAQ you can find this post which has more detailed description than help center: What are the badges I can earn on each site, and what are the exact criteria for earning each badge?
For more detailed description of the criteria for asking badges (Curious, Inquisitive and Socratic) you can read he description in this post: Asking days badges. I will also point out that the wording was updated after this discussion: Misleading description of new Curious badge. (I found both those links from a question which is very similar to yours - just posted on the main meta: Definition of a 'well received question' for the purposes of the Curious badge.)
Since Illuminator badge was mentioned in the comments, I will add a link to a related discussion on another meta: Refiner and Illuminator badges confusing. The badges Explainer and Refiner. Clearly, the description doesn't mean that you have to answer and edit 500 question within twelve hours. If on a single question you do those two actions (post answer that reaches positive score and also edited the question) and these two actions are within 12 hours of each other, the counter describing your progress increases by one. (As already mentioned, you can check your progress towards those badges using badge tracker in your profile - unless you already have that badge.)
1 In general, FAQ on meta are easier to update than help center. So this is probably better way for information that might change often. (Just as an example I will mention that in the part of the help center "What kind of questions should I not ask here?" there are several clearly outdated links. For example, neither Ask Dr. Math nor NRICH can be now used to ask question, yet they are still listed in the help center. I would hope that a corresponding post on meta would be updated more promptly in such cases, since it can be edited by any user with sufficient reputation.)
2 The tag-info should be used for guidance when to use the tag. But probably it might be good idea also to store there the most relevant links related to the specific topic described by the given tag. To name just one example, when I mention Data Explorer here on meta I typically include the link to tag-info. This has the advantage that users reading the post have quick access to some basic information - and if some of that information changes it suffices to update it in a single place (in the tag-info) and not in every question.