Can you shower with it?

The question has an answer less obvious than you might think. The short answer is: YES. The Long answer is: if you're showering with a low pressure water source, then no problem, BUT if your are showering with water flowing through a naked pipe, as it is common place in some parts of the world, the stream of water can impact the wrist with a whopping 13-15 bar, believe it or not, mostly due to potential energy as the water falls from above. Casio watches usually resist double the pressure it says it resist, however that capacity might fade with time, i.e rubber gasket degradation. So you can definitely damage a watch with a rating of "100m" water resistance showering in a developing country (read rural south america/central america), and can definety damage it if you are below a great fall of natural water, say iguazú, or Niagara falls. so it's nothing to do with depth as it is with pressure, and that's related with total impact energy.

Does the watch have a vibration alarm?

No it does not. It's got like 5 different alarms though. Even if it doesn't vibrate, this is the best watch I have ever owned, hands down.

Does it have 24-hr time? This is critical to my work.

Yes, it has 24 hr. time. My son loves his watch.

Does the watch give latitude and longitude?

Previous answer is correct; long/lat can be input to give you the most accurate sunrise/sunset times, but no display capabilities to show what your current longitude or latitude actually is.