I have an S40 1.9T,
Whenever you start the engine, the motor turns for about 3-5 seconds then starts roughly, occasionally dying out, and has a cyl 3 misfire, which causes the check engine light to come on. Unless you throttle it up, after 2-3 seconds of the rough idle the car starts fine, and doesn't get a cylinder-3 misfire after the car starts idling right.

given this description, can anyone maybe even make a guess on what I should be looking for that's causing this problem?

 

I would look at the 3rd cylinder coil. My 01 s40 has 28k miles and does the same thing.

 

Sounds like a common fuel pressure regulator leaking into the intake. The FPR should be under the throttle cover. Remove the vacuum hose from the regulator and see if it is wet with fuel.

 

My 2001 1.9T also had rough starting. It was a faulty fuel pressure regulator (not expensive). I also had misfire issue, but it did not coincide with the FPR problem (it was due to a clogged fuel filter). Anyway, I would start with the FPR and see if it resolves both issues.