- Can we post answers that are very similar to an older answer, but updated to be shorter?
- Can we post answers which would be better suited as "Saved n bytes" comments?
Yes, of course! Trying to out-score the other players is the whole point of Code Golf. If you can take inspiration from their attempt but add your own "twist" to make it shorter, then you are well within your rights to post a new answer.
Obviously, you must follow our attribution requirements, giving explicit credit to the author of the original answer and including a link back to that answer. But the CC BY-SA license explicitly allows adaptations and remixes, so this should be encouraged behavior.
- Do the answers to these matter/change depending on how active the user is?
If you feel that your change is relatively trivial and/or you want to give the original contributor a chance to update their answer, you can choose to do so. We don't need an official policy on this, though. Users can decide for themselves whether they want to post a new answer or leave a comment on an existing one. They can also establish their own thresholds for this, whether based on significance of the change, recent activity of the original contributor, etc. From the site's perspective, those don't matter.
- Does that fact that some improvements are only possible due to language updates matter?
Don't we already have a policy covering that—something like, you have to use the language as it was implemented at the time of announcing the challenge? If not, I think that's a reasonable rule.
Although I wouldn't have any objections to someone posting a new answer that takes advantage of new language features, I think that should always be a new answer, as it should also be treated differently for comparison/scoring purposes.
I know golfing languages are updated frequently, due to their niche purposes, but with mainstream languages, this is already conventional. A C++17 answer is a different entry than a C++11 answer—for the purposes of code golf, they're essentially different languages. I'd say the same would be true if you were exploiting new features added to a language since the challenge. I see no cause to discourage users from doing so, but these submissions become new entries in what is a de facto different language and should not be compared/scored against the original submission.