[Schmee]

Is there a log that shows open and close times & dates for all applications?

[Dair Grant]

There isn't a system log file which shows this information, but if you have an application you can register for kEventAppLaunched/kEventAppTerminated Carbon Events to find out when other apps are launched/quit.

This will only cover GUI applications, if you want notifications for command-line apps you need to use a Mach "port".