Heat gets warm then cold

Check coolant level. Start cold with cap off and add coolant as necessary with HEATER ON, while running. Let it get up to operating temperature and rad fan comes on 2 or 3 times, then make sure topped up and put cap back on.

LET YOUR CAR WARM UP FIRST!! or you risk the chance of over heating and casuing engine damage. start your car let it warm up for 5 to ten minutes and then try the heat. also when you start you should NEVER have heat on right away. the heat that you feel comes from the engine. and if the engine is cold and you are trying to heat it up it will put stress on the engine and over heat. ALL cars have to wait for their heat. that's why you see peoples car running in the morning with no one in it. so when they do get in it they have heat. heat will not work until the engine is warm.

Is your eng get warm if you eng is not heating up it can be a thremaste stuck open if the eng is heating up ck the heater hoses if the top hose is hot and the bottom is cold the heater core is pluged if the two are hot and warm the blend door is not working

See if your library has a Stratus shop manual and check the heating system chapter. You didn't say if the engine was reaching operating temperature or not. If the engine is warm but you are not, I would say the problem is in the heat exchanger somewhere. If your engine is cold, you can't really expect the heater to get warm, now can you. Then you have a different problem to chase.

Lack of water on your floor means your heater core is not leaking. It could still be plugged somewhere. The fact that you get some heat initially, but it turns cold when you turn the fan on, sounds like a lack of adequate warm water to the heater core to me. If the engine is warm, it could be a circulation problem of some sort.

Ok well every heating system and A/c system work very much the same...there are some troubleshooting Steps I would suggest...
First for the heating issue:
Step 1: Check Coolant level Low coolant can cause warm to cool air threw your heating system, not enough to fully heat...Caution: make sure the engine is cool when checking..You should have a tank that you can see the level..if not the top of the radiator is good.
Step 2: Any leaks in the coolant system, by just looking under the hood depending on the color, (Purple, Red, Orange, and Green) The system needs to obtain a certain Pressure to be effective..
Step 3: If all above checks ok...Assuming the blower motor is working correctly and good flow is present, there may be a vacuum leak or electronic divert-er malfunctioning( mixing cold and hot air thus getting warm air)
Step 4: Bring it to a shop, you may have a plugged heater core, an malfunctioning Thermostat, or an inefficient water pump...Hope this helps.

As for the A/c Not cooling...All cars that are 96 or newer must be charged with R-134a refrigerant...This can only be serviced by a shop that can deal with A/C recharging...could be low refrigerant, bad a/c clutch, bad compressor good luck hope this help

To check if the thermostat is bad, start a cold engine, and feel both of the radiator hoses.
While they should both start cold, only one should heat up slowly. The other should stay cold.
Then after the warming one gets hot, only then should the other hose also suddenly get hot.
If both slowly get warm, then the thermostat needs to be replaced.
(By the way, make sure the coolant is full.)

flush the colant

Have you checked the coolant level? If it's low, it will result in diminished heat, and could be the problem.

the blendor actuators strip out the gear inside the actuator cases on these all the time. replace the actuator.

air trapped in cooling system. loosen radiator cap (cold engine), fill with coolant. run engine with cap loose till warm to purge air. add coolant. repeat if nec.