Newbie Needs Help

Hi everyone!
Just got a used 08 2dr hatch. I have a code reader on the way, so I haven't been able to check her codes yet. She sat for 3 years with what was thought a cracked block. Turned out it was some sort of a broken pipe/cooling hose...It was resurrected and had its water pump replaced, and timing chain reworked, etc. It was then moved back to owners home, where it sat for another year. That's where I came in. We've been driving it for the last 2 weeks. It has kept a check engine light on steadily, but it has run perfectly, with no issues. I was told it had that problem but always did run fine. Everytime it was cleared, it would come back shortly afterwards. I assumed we would run into some issues with it being unused for some time. As soon as the reader gets here, I expect to be able to get some of these little niggles sorted.
Just a couple days ago the temp light came on, and then went off. The next day it came on yellow, went off, came on red a couple minutes later. It went out. Checked coolant, looked for puddles, no problems found. Next day, again after 15 -20min., light came on and stayed. This time the fan stayed on 2-3min. after engine was turned off. To me this sounds like a sticky thermostat. Any other car I've owned, it's understood, when you change the water pump, you always change the thermostat. On the mini, I'm not sure if that's a standard procedure.
My question is; do you think this is a thermostat issue? If not, what do you think I should look for?? I love this car and I want to get her straightened out and keep her for a long time! Thank you to anyone who can help:big_grin: