how/where can I see what happened when my Android application crashes in Eclipse (using Run, not Debug)?
Sometimes the device gets "out of sync" with eclipse and logcat doesn't show any messages, as you've discovered.
To fix this, try a) closing the logcat tab and creating a new one; b) disconnecting your device and reconnecting it; c) exiting eclipse and restarting it; d) rebooting your device; or e) rebooting your computer, in that order. Usually the problem is fixed by the time you've done c).
I found Eclipse not very reliable and switched to IntelliJ, much happier that way. IntelliJ 9 supports Android development (not the community edition though). Beside Eclipse DDMS, you can also see the logcat from the command line. Open a shell and change to the folder of your Android SDK, and here to subfolder 'tools'. There you can run
adb shell logcat
which will show you the log.
If you want to explicit distinguish between real device or emulator (in case you're using both), you can add parameters -d or -e
Device:
adb -d logcat
Emulator:
adb -e logcat