To 1, yes, that happens. Makes it impossible to have cities where not every area is connected by roads. It's also why it is so important in this game to not have traffic jams. There is a recent mod for this however: https://steamcommunity.com/sharedfiles/filedetails/?id=16808..., Most Effective Traffic Manager. It changes how the routing works and tries to combat exactly this. Though I could not just add it to my current rather large city, it dies with null pointer exceptions. It might work better now that it got some patches or when starting from scratch.

The "too close is bad" is done by noise pollution. I think hospitals do not emit noise in Skylines, but the cars going to and from do, and other buildings have that built in, especially monuments, malls and metro stations.