If the battery has been problematic lately, you may want to look into your running processes and seeing if anything has been taking up more CPU usage or memory than previous. You can also try using software like BatteryBar to determine your battery's wear levels:
http
/batterybarpro.com/
As for your thermal throttling incident, it's not very likely that you permanently damaged your machine - the thermal throttling is in place to prevent such damage - but you could possibly have ruined your thermal paste, or caused melting of a plastic part in some rare cases.
If the crash means that the PC can no longer turn on, then you should have it serviced. If the crash occurs only occasionally and allows continued use, then it's possible that your cooling system is still tripping the thermal throttling, and you should have the laptop cleaned and thermal paste reapplied.