2006 Chevy Monte Carlo temp. Gauge says running hot, but th

Ok so im having a few problems, that all seem to be related. First my a/c and heater both are not working properly during idle and only work well when I reach high rpm. Second, occasionally after turning my car off, ive been noticing that my fan continues running for 5-10 minutes. And lastly my engine temp. Gauge has been moving up and twice last week gave me a message to turn engine off. Other times it just returned back to regualr temp. On its own. About 6 mos. Ago I had similar problem with temperature gauge and brought it to a lube shop to check. They told me my car had a "bad code" and said they reset it for me. Now its doing same thing and im soooo lost of what it could be. Im trying to avoid spending unnecessary money.