It sounds like your temp gauge is rising when you stop and falling when you are stepping on the gas. If that's the case that could mean air in the cooling system. To bleed the air you need to run the engine with the radiator cap off and heater(s) set to high. You should see the coolant bubble at the radiator neck and the level should go down. Do this at least until the engine gets to normal operating temperature then top off the radiator. If the '05 has a bleeder on the thermostat housing it may be easier to do it that way, but I've never worked on a car with that feature so I never learned how to do it.
As for your no-start issue, will the car start with a jump start? If so I'd suspect the battery but it could also be the alternator. If it still doesn't start you could have a problem with a number of other things like the starter, ignition coil(s), alternator, etc. I find it unlikely (but not impossible) that the engine has seized up - there should have been other symptoms like noises or vibration/shaking.