Perhaps something to bring up on the forums to see if anybody has input on how to do this! I personally haven't had such a bad case locally yet. But you're having a great impact updating it as most likely people pass up on using OSM based services because of it, and by having it up to date it may encourage those people to switch in the future.
I'd personally use the web editor at https://www.openstreetmap.org/
Make an account, go to the zone you want to edit and press the edit button. Add a line and once the line is added you can add a type to it (you can check the wiki if you don't know which one to use)
You can enable tracking on streetcomplete that will track your walk/bike/whatever and you can use that as a guide on the web app should it be hard via satellite (do note that this tracking is posted publicly).
I personally turned off the road width data as I always have to cross the roads and it both feels awkward and is just not that practical imo. But you can use any other measuring app if any exist to get the width and manually put in.
Tomtom Amigo also uses traffic data! Although their dataset is ofc smaller than google maps as less people use it. (Still not FOSS, but at least not google)
Unsure, in theory it should be possible as long as the data is on OSM. I do know Mapy.cz is regarded as a great map software although idk how the navigation/biking part. I personally use Mapy.cz for anything not car related (but barely bike atm and haven't needed walking directions as of yet) and tomtom amigo for car navigation.
Enjoy and thanks for contributing!