my laptop has taken to closing down without warning

Try this, it has worked for us. Unsure on the battery advice though, waiting for clarification on that.

BobRz

Reset the SMC. I'm pretty sure that model has a removeable battery, so you want this method:

Resetting the SMC on Mac portables with a battery you can remove

Note: Learn about removing the battery on MacBook and MacBook Pro.

1. Shut down the computer.
2. Disconnect the MagSafe power adapter from the computer, if it's connected.
3. Remove the battery.
4. Press and hold the power button for 5 seconds.
5. Release the power button.
6. Reconnect the battery and MagSafe power adapter.
7. Press the power button to turn on the computer.

You shouldn't be running the battery down that low on a regular basis though. It'll shorten the life of the battery.