Popeye, far as I can tell, the time shown is the time at your home port. In other words, as you cross time zones your boat's clock is not updated to the new time zone. So the farther west you go, the later the sun rises (and the later it sets) as far as your clock is concerned.
It's no big deal really to tell when sunrise and sunset is, if nothing is happening I go to map view and run up the TC. Say it's night - the instant the map changes from "night" mode to "day" mode I go back to 1x TC and have a look. Well guess what, the sun is just starting to peek over the horizon.
Same during the day - when the map changes from "day" to "night" mode, well the sun is just starting to set.
Of course if stuff is happening and I'm running at 1x TC it's easy enough to see for myself (from bridge or periscope) if the sun is rising or setting!
Regarding weather report, like -Pv- said, the devs could have left it out completely. Just be glad it's there. And it's really not much harder to implement than clicking on the button that really should (but isn't) there
DH