Probably bicycle=yes. It has to be filled in though, not just left on the default yes, otherwise, the data isn't picked up.

Also, I do feel the bridges should be marked "designated" since it is mode separated and there is specific signage: https://photos.app.goo.gl/c3izAdKiGUinc4AT6
https://photos.app.goo.gl/SXCkcwqwWmyeUUkg6

What's the reason for the section of the multi-use trail between Mill and Park having the tag "bike=yes" while the rest of the trail uses "bike=designated"?

well, RideWithGPS, for example, uses OSM data to give directions to cyclists, and it prioritizes routes with bikes lanes or bikes paths. Putting in the wrong data could put cyclist on a dangerous street or put them on a route where they are actually not allowed to go.

I suppose you could put "not specified" or just leave it blank.

The problem with putting bike=yes on sidewalks is that they show up as bike paths on the OSM Cycle Map, and it will also be used by directional apps and put people on sidewalks when they really shouldn't. The blue implies that it is a general purpose bike trail.

Are bike allowed on these sidewalks? As far I know, bikes are only allowed on the path on the south side of the street Illinois and Wall, at least that's what the map indicates. I know it is signed on that portion as well, but not this portion.

Ok. I suppose you're right. one of the exceptions, however, is the path on the south side of grand ave. There is a sign that specifically states that it is for bicycles only and that pedestrians are prohibited. This is also supported by the official carbondale bike map https://explorecarbondale.com/DocumentCenter/View/2707/City-of-Carbondale-Bikeway-Map-PDF