Having two separate gauge issues with my 2000 Chevy Lumina. Are they related?

The first issue is that my gas gauge, when I’m low on gas (we’re talking like 1/8th of a tank on bad days), while I’m driving, I’ll notice the pin swings all the way to ½ a tank or over, and then slowly drops again. It’s not because I’m on a hill or a dip. I’ve watched it while I’m sitting on a flat roadway at a red light, and I’ve watched it while driving straight on a highway.

The second is my temp gauge. The pin doesn’t go up at all, meaning, it always shows that my car is cold and not warmed up, even after I’ve been driving a while. In relation to this, it seems like the coolant light keeps coming on even when I have coolant, and the heat doesn’t really get very warm. Sometimes it blows cold if I’m not pressing the gas pedal down.

Any ideas?