Misfires when cold

white smoke when starting from cold is
condensation of moisture in the muffler/ tail pipe
that is normal .... till the engine warms up

heaps of white smoke when hot
suggests coolant/steam in the combustion chamber

blue smoke is oil
black smoke is fuel

Your will need a road test with a professional scan tool
& record & look at data

IF ---you never do any Preventative Maintenance & few folks
do,anything at all,then problems are coming your way

Flashing MIL Lamp-- is a Class A-- Exhaust Converter Ruining
Cylinder Misfire

Runs Rough,Black Smoke --BUT--you keep using it & only
now, think maybe something is wrong !!

That car has lots wrong,that needs work,way too much to
write a book on it here, for you

Your Fuel Trim must be sky high

No fuel control & lots of misfire counts

The 14 years from 1997 to now, the converter is worn out,
but you have lots to do before replacing that

Starts good cold, because your in open loop & the fuel is
controlled by the PCM

Once it warms up & goes into closed loop,you have no
fuel control

What are you asking ?

Sounds like the motor is gone mechanically

misfire is bad timing
check cam time.
check spark timing.
bad timing causes backfiring..

did you put the spark wires back like you found them or better the correct firing order? if not it backfires.
last is lean backfire.

i would look at the temp sensor. i believe when they fail, they peg out all the way cold or all the way hot. so your truck thinks it is still cold when you have reached operating temps, causeing your misfire.

Circuit malfunction is often caused by damaged wiring. Also check the pins and terminals at the connector carefully.

A 2008 mercury what? What is the model and engine size?

I’m happy to assist further over the phone at https://www.6ya.com/expert/kim_3202917d7c587455

p 0300 is random misfires and 0302 is no 2 cylinder misfire
most misfires are from dirty injectors rather than plugs
run a quality injector cleaner and see if there is improvement

Without your actual trouble code, I am assuming that you are have a misfire on cylinder # 5. The computer cannot really tell what is causing the misfire only that cyl 5 is not putting out full power. If it misfires when cold then smooths out as it warms up, I would say it is probably an injector that is sticky, as it warms up it starts working properly.

Some smell is normal if it has been quite sometime between oil changes. This happens more with short distance driving.

How does it get there?

Through cylinder blow by when the engine is cold. The piston rings leak minuscule amounts into the oil which add up over time.

If you are having no other running issues (IE black/drk grey smoke out tail pipe with oil residue on pipe tip) I would not be alarmed by this.