Re footway: of course, not tagging trottoirs as highway=footway but instead as highway=path would be absurd. But that’s because by law trottoirs are both reserved to pedestrians unless otherwise signed and mandatory for pedestrians to use (Art. 43 SVG). That is to say, their default rules are exactly the same as if they were signed with the blue footpath sign. We can and should take this into account when tagging.
However footpaths that aren’t along roads and so aren’t trottoirs don’t have the same legal defaults and really should be tagged as highway=path unless there’s some sign contradicting this, because otherwise the default access rules of highway=footway would be bicycle=no horse=no, which isn’t supported by the evidence.
In the reverse case, it is very common for the bicycle pictogram to be used in areas where bicycles are only permitted, not designated. Therefore just seeing the pictogram should not immediately make us jump to bicycle=designated (perhaps co-equal with foot=designated). Unless there’s good evidence, I think it’s better to assume an implied “velos gestattet” sign, and tag bicycle=yes.
Good evidence can include e.g. the situation at the Stadttunnel, where the tunnel is prohibited both for motor vehicles and for pedestrians. This leaves by process of elimination only bikes and horses, and horses in the middle of the city in this day and age is obviously not the designated users of the tunnel. It also should include any separated bicycle lane, like the one you linked on Zweierstrasse, which is taggable as cycleway=track, since bike lanes/tracks are usually only marked with paint and blue signs are not typical (though they are possible, as on Birmensdorferstrasse).
Bad evidence is IMO, e.g., the one you linked at the Schmiede Wiedikon intersection. This is legally part of the trottoir, so the designated user is pedestrians. At most I can see an implied “Velos gestattet” here, since there is no signage and not even separation by paint/curb from the rest of the trottoir, unlike further down on Zweierstrasse. Tagging bicycle=designated implies that bicycles and pedestrians are equal here, but they aren’t.
Similarly, on Bärenbrüggli, tagging both as designated because they both have a pictogram painted implies this is a shared cycleway, since that’s how those are tagged too. But there are differences between a road closed to motor vehicles (which is how it’s actually signed) and a shared pedestrian/bicycle path, and I think our tagging should try to capture that difference. I think this case is more debatable than the trottoir case, though, as there is no clear designated user legally.
I know in Germany they went for a different tagging solution, where they tag which sign (if any) the path has directly. However this is not very well supported by OSM data consumers.