It sounds as though you may have a leaking injector. If so, it would be flooding the engine after a period of time, causing the hard start and the slow resumption of running well when it does start.
The easiest way to check my theory is to put a fuel pressure gauge on the fuel rail over the same period of time. Check the pressure after attached, then again in the several hours after when you'd normally see the issue. If it has dropped significantly or all the way to zero, there's a problem in this arena. To further diagnose, you'd then pull up the fuel rail with the injectors attached. Turn the ignition to on (do not start), and check to see if any of the injectors are leaking. You'd want to do this into some sort of container so as to not cause a fuel spill and potential fire hazard.
If it does turn out to be a fuel injector, there are some great videos on YouTube which can show you how to clean them, or take them to a shop and have them do it (at a much greater expense).
Obviously, if the fuel pressure stays up to where it started (as it should), this is not the issue.