13" MacBook Pro 2009 Spontaneously Shuts Down

Question

For the past few days, my MacBook has been shutting down without warning while plugged in and fully charged. I initially thought it was due to an old battery, since the battery life had been unpredictable enough to cause sudden shut downs at around ~15% battery life in the past, so I had the battery replaced today. However, it has not fixed the sudden shut down problem. I've reset the NVRAM and SMC, since I read it could help the problem, but neither has helped.
From what I can tell, there is no common factor between the shut downs. Sometimes they occur shortly after one another (for example, sometimes I'll turn the computer back on and it shuts down again before I can even open a browser), and sometimes they occur hours after the previous shut down. Temperature doesn't seem to be a factor, since the laptop has been quiet and cool almost every time this has happened. Also, every time the computer restarts, the clock is reset to ~2000, and I have to manually change it to 2013 before selecting a working wireless network. (However, once I'm connected to the network, the time does successfully sync with the official Apple time, as per the default setting.) The only other detail I can think to add is in regards to the power cord's indicator light: Whenever the laptop suddenly shuts down, the light goes dark. Also, even if I try to turn the computer back on while the indicator light is dark, I'm unsuccessful. I'm only able to boot the computer up once the indicator light returns. (99% of these random shut downs have occurred while the charger was plugged in, so I don't have much information for how it behaves on battery power. I'm currently testing that, but because of the unpredictability of the shut downs, I don't know when it will happen again.)
Is there any precedence for this? I'm sure I need to take my laptop in to an Apple store to get it checked, but it would be nice to have an idea of what's wrong with it—for example, what could be wrong/damaged and a ballpark figure of how much it could cost to fix.
Thanks!
Laptop information:
Model Identifier:          MacBookPro5,5
  Processor Name:          Intel Core 2 Duo
  Processor Speed:          2.53 GHz
  Number Of Processors:          1
  Total Number Of Cores:          2
  L2 Cache:          3 MB
  Memory:          4 GB
  Bus Speed:          1.07 GHz
  Boot ROM Version:          MBP55.00AC.B03
  SMC Version (system):          1.47f2
Charge Information:
  Charge remaining (mAh):          4792
  Fully charged:          No
  Charging:          No
  Full charge capacity (mAh):          5854
  Health Information:
  Cycle count:          0
  Condition:          Normal

Answer 1

There's really no telling what's exactly wrong with your machine without having the diagnostic tools to check it out - which Apple techs at Apple Stores and AASPs have. So my advic eis to get it diagnosed and get an estimate for repair.
There are just some things that need 'hands on' diagnostics that are impossible to do over the Internet.
Good luck,
Clinton