Dumb question. But have you checked the entire charging system and battery to ensure they are working properly? Couldn't tell from the pictures you posted what the age/manufacture date of the battery was.
The battery shouldn't be dying or draining fast if the Corolla is left for long periods of time (two or more weeks). As always check to ensure nothing (like radio, third party security system, light, etc.) is drawing constant power/current. Could be an old battery (or dead cells) combined with a alternator that isn't producing the proper output. Or short somewhere causing power drain.
Couple of months ago, left an 05 Corolla LE with an 8 year old battery sit for a month and a half without running and it started up fine. Because I currently work from home in a city that same car often isn't driven much and when it is it's for 5 to 10 minutes often every other week. No problems starting the car (knock on wood).
