Note that the pilot screw is actually the idle mixture adjustment, and it has only a small effect anyway. If you have idle problems, it may be worth investigating, but it has nothing to do with cold starting problems.
If the air filter will not slide out easily on the carburetor mounting studs when you remove the two nuts, it sounds as if there is a retaining screw, which could be below the body of the air filter, toward the right from those two mounting studs, or perhaps inside the air filter housing, extending through it.
Here is the short-form workshop manual:
http://www.honda-engines-eu.com/en/images/114198.pdfIf you check the movement of the Choke Rod and the Choke Joint Lever when you put the speed control lever into the choke position, you can see how far they move. You should also be able to flex the Choke Rod slightly without bending it, to see if the choke lever on the top of the carburetor is capable of moving further when the speed control lever is against its stop.