Illustrations are used in figures. I'm not sure I get the issue of this point. Are you saying figures can't be decorational?
Hm, decorations are a tricky one, but for weather these kinds of images are usually considered content or like an icon. I believe they could be considered a figure. Figures are pretty much any non-text content and if it has a corresponding text component then it can get a figcaption. The idea of figcaption isn't much more than, "how can we state with HTML semantics that this bit of text is, specifically, describing or relating to this image/video/whatever"? For figures they've made what it refers to pretty broad: charts, graphs, videos, carousels, slides, images, etc. So I'd say if you want a "caption" for your image then figure would be appropriate. I don't see the point yet today of making any content image that's just by itself a figure though... maybe once it means something to user agents?
No, it's not a list of anything.
I used a list because it seemed to be a "bullet points" set of statements. They're not complete sentences and they have a grouping topic so I tried a list. I would not be against something else being used.
<aside><b>34°F</b><br />Wind chill: 29°F</aside>
This still makes no sense to me. It had better make perfect sense when it's all by itself, since the idea of an aside is that it can be completely separate and viewed without the rest of the content around it and still make sense on its own. The kinds of sidebars people tend to make into <aside>s now would be things like "Blog archive (list of blogs)", "3rd-party ads (list of ads)", etc. I mean, you don't need to read the current blog or whatever on a page to view the archive on its own. That aside would have a header in it (<h?>Blog Archive</h?>) and whatever other structure necessary.
I don't believe "34°F Wind chill: 29°F" by itself can be separated that much. It could be "<h?>Weather for [city], [date]</h?> Temperature: 34°F; Wind chill: 29°F" though. That bit could do well on its own and all the other information would just be additional and not necessary (which seems to be the reason you wanted to use the aside).