So I talked to the cook, he ran away and I couldn't find him. When McCoy went back home, the cook showed up and began to attack. I couldn't search the body after. McCoy kept saying yuck. I know there's a clue on the cook's body and it seems that there's no way to get it doing this pathway. Does anyone have any problems with this?

I'll have to reload and confront the cook straight away after he tossed the pot at McCoy. I hope I'll get it.

Yes, this is by design as far as we can tell.
There are a few things that won't happen if the cook (Zuben) ambushes McCoy on the roof of his apartment.
One of them is that he won't search the body afterwards.

Somehow I cannot "trigger" Zuben to drop from above at the locked door ... and any scene there afterwards. So I have no dialogue with Gaff. Only when I fly home (now actually my random choice instead of story/dialogue driven) Zuben attacks me on the roof.

Reload does not solve it. The scene plays out always the same way. What goes wrong here in the door scene? How can I "force" trigger Zuben to drop from above?

Thank you very much, that helped. Now the story flows smoothly before my eyes ...

You have to be quick in general.

Dodge the pot, follow Zuben, ignore the tramp (he'll just tell you to go inside, so it's a waste of time and money), head straight for the stairs and click on the locked door.

Unlike the rooftop outcome, you can actually choose not to shoot Zuben here, just holster your gun.

Be careful: you can also skip Gaff's scene if you don't go straight towards the police lights afterwards, but it's no biggie as Crystal gives you similar info in Act 2.