There's a lot here, but Superman as a character has a number of problems in film. His powers are often overclocked in media. It's generally accepted that Superman cannot outrace the Flash and is not capable of light speed travel. If Metropolis were New York and Gotham Chicago, I would expect Superman to take at least a half hour to get there.
Also, with regards to Batman finding Superman's mother: Batman is, above all things, the smartest of the Justice League members. He would recognize Superman immediately and devise a way to make that happen.
Yeah, didn't it start out he couldn't even fly and thats where the whole leap over a building in a single bound came from? I know his ability to fly was added later.
Haha I like the comparison of city distances I'm not sure, but if anything I always thought Gotham and Metropolis were like LA/NYC as far as distance.