Engine Overheating

The black smoke is likely due to a rich fuel/air mixture (too much gas is burned).

You could have a leaky fuel injector, a defective fuel pressure regulator, a clogged air filter, a problem with the MAP sensor, a defective coolant sensor, a defective o2 sensor, TPS sensor, problem in the EGR system.
You could also have more serious internal engine problems (like burnt exhaust valves).

First thing I would do: hook up an autoscanner to your PCM and check for DTCs/pending DTCs. Make sure the readings from all the sensors are within range.

Honestly, I'd be very careful with a car shop that starts replacing one part after another without doing any real diagnostic.
I suggest you take it to a different car shop (one recommended by a friend or something).

Given your description of the problem, a glogged catalytic converter could be the culprit indeed; however it could be many other things as well. (Check intake vacuum at idle; if low and keep dropping check your exhaust system).
If it turns out to be the converter, I recommend you replace (or at least inspect carefully) your O2 sensors. They are very important for proper operation of the catalytic converter.

If the converter is alright and the sensors test fine, I would probably take a look at the charging system (if you have a problem in your charging system, it would eventually result in your car turning off).
When your car turned off, were you able to start it right back up, or did you have to jump start it?

Also, do a fuel pressure test; it should tell you if there's any problem with your fuel system.

Check for vacuum leaks. It could cause your car to run roughly or stall. This should be more noticeable at idle, thus my following questions:

does your car turn off at idle or when you're driving?

Finally, if you still haven't found the problem, do a compression test; it will tell you if you have internal engine problems.

Neither f those would effect overheating and head gasket doesn't have to smoke to be bad they leak internally. Do what I said in first reply, radiator can have good flow and still be bad if you feel cold spots then it's no good shold be same temp all over.

No start?After it overheated
check for compression
check for spark check for fuel pressure
do a code check

You will have to pressure test the cooling system and track the leak back to it's origin.

$1,300 is absolutely ridiculous! Run from this mechanic and don't look back! (An honest quote would've been around $700)

A used motor with about 30,000 miles for this vehicle should run you around $700-900. The labor for an install will set you back about $500. As long as you go to a reputable (and honest) shop, they'll give you a 12 month/12 thousand mile warranty. That's a much better option than replacing a head gasket on a motor that has 160K miles on it.

No matter how far you can get the price down for a head gasket swap, DON'T BOTHER. That motor is ready to poop out anyway, especially after driving it 400 miles while it's overheating. (DON'T DO THAT ON THE NEXT MOTOR!)

And to find a motor, check on-line. Just google motors in your area. You'll find one, and when you do, ask them to recommend a mechanic to do the install.

You replaced the two parts least likely to cause overheating. If you're having to add coolant have the engine tested for a leaking head gasket. Your mechanic will use a glass cylinder with two chambers partially filled with a special dark blue liquid. He will draw air through it from the radiator while the engine is running. If the cylinder head gasket is leaking, the liquid will turn bright yellow.

Do you need to keep adding coolant?
Are the cooling fans working correctly?

Try bleeding it see below

Always bleed air from cooling system after replacing coolant. Set heater for maximum heat. Remove radiator cap. Loosen drain plug and remove drain bolt (if equipped) from engine block. Drain coolant reservoir. Fill coolant reservoir to MAX mark with 50/50 water-coolant mixture. Loosen bleed bolt and fill radiator up to base of filler neck. Close bleed bolt when coolant flows out without bubbles. Tighten bleed bolt. With radiator cap removed, start and operate engine to normal operating temperature. Add coolant if necessary and check for leaks.

If you coolant level does not go down then it is probably the condensation from the air conditioner working which will drip on pass side through drain hose. Otherwise have cooling system pressure checked.

I vote thermostat