are normal bikes allowed on sidewalks and walking paths

In California, yes, that's the default rule. Municipalities have limited authority to regulate road traffic, but can prohibit bikes from sidewalks if they post certain signs. As a result, most municipalities here allow it, with some downtown areas having restrictions where sidewalks are in heavy use.

pedestrian-only things

Sadly, the pre-1900 American conception of the sidewalk as a pedestrian refuge -- paved, free of mud and puddles, in an era where unpaved roads and horse carriages were the norm -- was entirely sidelined in the automobile industry's desire to move product. It would be great if municipalities actually maintained their sidewalks -- in the Western USA, most states make it the responsibility of the adjacent landowner -- but seeing as they're not even required to provide sidewalks, it's truly a block-by-block situation.

As a result, there's every incentive to reuse sidewalks for anything that can use them, despite the obvious walkability problems, issues for people that must use the sidewalk (eg wheelchair users), while also eating the risk of sidewalks not being designed for any wheeled device (and sometimes having no regard for sight lines that would give pedestrians a chance to see an automobile running a stop sign). The very existence of sidewalks in some places has been used to justify not building additional infrastructure for separate uses, such as bike lanes, based on the false premise "nobody bikes, so no need to build infra", which is obviously why nobody bikes. Some places only have a disconnected sidewalk segment because a building (eg school) required it for ADA compliance.

Note that a sidewalk can be properly upgraded to a multi-use path (MUP) that can support mixed traffic (eg bikes, class 1/2 ebikes, (e)scooters, dog walkers, strollers, and pedestrians) but these have to be much wiser (>3 meters) and are only designed for low traffic volumes. MUPs easily reach their design volume if, say, a school has ended class and all the students are walking home.

For the narrowest of legacy sidewalks (ones which would fail the ADA requirements today but are grandfathered if they're left unchanged since installation pre-1990) of maybe 1 meter wide, you are correct that there's no way for a pedestrian and bicyclist to fit in this space. The common result is that the pedestrian steps off and the bicyclist gets by. Under the law, it's supposed to be the other way around, and IMO, it's a lot easier for bikes to ride over grass or surfaces adjacent to the sidewalk.

I agree that pedestrians should have first dibs on sidewalks. The present reality is that this is all artificial scarcity: there's plenty of road space that should be rededicated from automobile lanes or parking so that sidewalks are widened and pedestrians and cyclists have the space they need. This simultaneously solves the e-scooter parking issue, reduces automobile congestion and driving speeds and collisions in cities, and improves air quality and civic life.

The modern automobile is quite good at traveling long distances. So there should be no genuine complaint with having cars drive the long way around a neighborhood or business district, so that pedestrians and cyclists can go the direct route to the front door. Through streets for cars don't need to exist; use movable bollards to allow emergency vehicles to drive into pedestrian areas