2003 sunfire, 136,000 miles: needs water pump. Should change timing chain, and all that goes with that. I have to pay for all of that & I wonder if the car is worth that because I know it's going to b

Yes change what easy to get to while change pump. Notice any leaks. Air in coolent system will cause temp gauge to fluctuate. Have you over heated the car?

No. Temp Guage will go from 95-110 to 250 in bank of eye. Let it sit for 5 min & it's down to 100. Only top hose is slightly warm & all hoses feel weak. No leaks though except overflow tan

If it's your daily driver, and it's been a reliable car otherwise, AND you think the transmission and torque converter are going to hold up for another 50K miles, and the tires are decent, then it might be worth it, rather then going back into debt on another new or used car. But if the body is shot, the tires are baloney skins, and the transmission has never been serviced, you might want to just put that 800 smackers this is going to cost you towards something new.

Try purging the cooling system from any air pockets..check this out to know how......https://youtu.be/nThsFGa1vuE