RT: If the idle speed slowly increases over 15 minutes (from a cold start?) that sounds like the action of the choke is a factor. The bimetallic coil inside the choke housing winds/unwinds with changes in temperature. When you press the gas pedal to the floor before starting a cold engine, that sprays fuel from the AP into the carb, deep down, and frees up the choke plate from the various other mechanisms that link to it; the only force acting on the choke plate then is that coil. Depending on ambient temperature, the plate should close a little or a lot. You had mentioned elsewhere working on that choke. It may need some more attention.
As soon as the engine starts, vacuum will operate a diaphragm and linkage to pull the choke plate somewhat open. There may be more than one device that allows or forces the choke to open.
The coil in the choke mechanism gets heated, by coolant in earlier trucks like my 1979, but probably electrically in yours. It coils or uncoils and puts a light force on the choke plate. In normal driving, pushing and releasing the gas pedal frees up the fast idle cam, which is one of the several devices that mechanically link to the choke mechanism. Or you can do a quick jab of the accelerator to free it up. That lets that light force from the coil open the choke plate fully. The cam, rotating to its warm-engine position, allows the main throttle plate to close to its normal warm-engine position.. so the RPMs, which are faster with cold engine, to revert to normal.
If after 15 minutes you kick the accelerator pedal, do the RPMs return to 600 or so?
Re: the link between the AP and the throttle - you can probably operate the AP by itself, via its lever. If you operate the throttle, the AP's lever will also operate.
Gas should not leak out of the AP, if by that you mean it's leaking outside. The AP should spray gas deep down in the main (smaller) barrel of the carb, where the only place it can go is into the intake manifold.
You're probably not able to see the carb well enough until you remove the air cleaner. I can't think why the engine won't idle with the air cleaner removed but it might be some vacuum hose you inadvertently pulled that needs to be kept in place or plugged.
Once you have a better view of the carb, can you verify it is the same one you posted an exploded diagram of in another post?