Year, Make, Model, : 2002 Honda Civic LX

Year, Make, Model, : 2002 Honda Civic LX
Kilometers: 280,000
Problem:
When I turn on the ignition key, it doesn't want to start. After a delay or during the second attempt to start, the engine starts.

Details of the work done:
1. When the mechanic checked the battery, it was half-dead. So, the battery was replaced. The problem remained.
2. So, as a next step, the spark plugs were replaced. The problem remained.
3. Again, the mechanic suggested starter motor, that was also replaced. The problem is still there.
* The battery, spark plugs and starter motor that were replaced are all original Honda parts.

Further details:
---When does the problem get worse?---
* If I drive less short distance, say less than 10 kilometers and shut the engine off, I will have trouble starting my car again even if it is within 5 minutes of shutting off the car
* The problem can get more severe when the temperature goes below zero degrees

---What do I do to minimize the problem?---
* If I pour a lot of fuel injection cleaner to the gas tank, the problem subsides. It doesn't go away.
* If I drive my car for 60 kilometers and then shut the engine off, the next time I start the engine it starts without issues.
* In the parked position, BEFORE turning off the engine, I push on the gas hard and turn off the engine (this lessens the starting trouble to a great extend)

Some more notes:
* When I turn on the iginition key and when the car starts, it sounds as if it is going to die/stop. But it doesn't.
* Once it starts running it is fine.
* Again, once it starts it DOESN'T dies when it is idling. The rpm needle stays steady (means the rpm needle doesn't go erratic).

I have been searching the internet for an answer. So, what could be the problem?
If it is IAC - where is the IAC located on 2002 Honda Civic LX?
If it is PCV - where is it located?

Is it something else?

Thank you,
Best Regards,
Li