Small 'eye' icon??
The "eye" icon appears when you have your backlight set to auto. that means when you move your wrist to look at your watch the light comes on automatically. I have found this to be a huge drain on the life of the battery. to turn it off hold down the light button (the grey button on the front with the red "g" on it) untill you hear it beep.
As for the watch dieing when the alarm goes off that is a good indication that you need a new battery.
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