The PD applies to unwarranted or unwanted intervention. If Starfleet is dragged into situations against their will by the locals, or if they are asked formally for aid, then the PD no longer applies.
Not really. Gowron requested aid during the civil war, under the terms of the alliance
, and still Starfleet refused, citing the PD.
Of course, Picard has a notoriously broad view of what the PD forbids him to do.
Gowron was not recognized by everyone in the empire as the legitimate leader of the Klingon Empire. Yes, Picard arbitrated for him but that Duras had enough supporters to challenge him in such a way shows that this wasn't what all the Klingons thought. If the Feds helped out someone who was not recognized at the legitimate leader by the people and more or less "installed" him it would ultimately damage Gowron's position and be seen as the Feds interfering in an internal dispute that the Klingon people did not wholly want them interfering in.