1989 mitsubishi galant, when engine is cold has severe miss, after reaching operating temperature miss disappears, and engine runs smooth.

Sounds like you've done everything but the injectors. Run a bottle of fuel injector cleaner in a tank full of gas every time you do an oil change after this time. Also, check the fuel pressure to see if it's where should be for the injectors. But then, if it has enough pressure to go through the dirty injectors to make it run, pressure should be good. Check it anyway.

run fault codes as this sounds like a solenoid/ servo problem that worsens when hot.

Although your temperature gauge may show that it works, some models have a second temperature sensor for the engine management system.
That second sensor provides info which sets curb idle and timing advance. The "Cold start" feature is a part of this. The hic-cup is something of a bucking symptom as the engine controls toggle between hot and cold run settings repeatedly as water temperature is not stable.
In the past, I found several 1984 engines having defective thermostats which are tied into "Step-up" solenoids for the curb idle. What happens is idle speed is suppose to drop when the engine is hot.
When someone adjusts the curb idle while the engine is on the "cold setting", the setting(rpm) will be too low once the engine has heated up and the part drops to a hot setting.
So to clearly define your problem, you have 2 stemming from the same defects. The "in motion" problem is caused by timing changes based on temperature and the "idle" problem is caused by improper adjustment of the curb idle solenoid because of unstable operating temperature.
What you want to do is change your thermostat. This will give you a stable temperature. Then let the engine heat up and you should notice a "click" and the engine should idle down. It may stall. Then turn the idle rpm up until the engine will not stall.
The hic-cups should disappear once the operating temp is reached. If you still have problems then look for the water temperature sensors before replacing expensive parts.

several reasons why an engine will miss. most common problem is the spark plugs and wires are bad.

check for vacuum leaks ---transmission actuator and hose
vent control hoses , any where the there is vacuum control
the idle air control sensor would be a suspect so check wiring , connections and the sensor itself

Could be the engine temperature sensor is bad, or just un-hooked.

bubbles on coolant extension is the pressure produces by the high temperature when the engine heats up. pressure always finds the nearest escape. need to change engine thermostat.

Any engine, when cold, takes at least 3 miles before reaching operating temperature.. After reaching operating temp it still takes a mile or 8(depending on your vehicle) before "over" temperatures are reached. Therefore if needed, a vehicle can be driven several miles without any coolant/water at all before overheating and causing damage, if started from from cold.

check engine coolant temperature when it gets warmer, or at operating temperature, and verify that the temperature gauge is showing warmed up to operating temperature. If the coolant temperature sensor tells the computer the engine is cold it will over fuel and miss and the engine will die. When cooled down it will run fine until it warms up. Also could be coil if it starts missing after warmed up.

you can try a tune up first or you can have your compression check you can have a crack head..it will keep an air bubbles in the water an mess with the temperature sensor. but you must be adding water to the radiator. do you have any water in your oil? if not a major tune up will fix you up.