Assume for a moment the Russian Government is entirely preoccupied by a internal crisis in the summer of 1914. So distracted the Austrian/Serbian affair cannot be considered. As I understand it this largely waives away any chance of German. France. & Britain weighing in & any Great War starting that summer or autum.
So, how does the Austrian/Serbian confrontation play out that year?
You pretty much need a revolution in Russia for that, not unlikely.
1. Is there any chance the Austrians will take military action because they can?
Well, if your neighbor orchestrates the murder of the heir to the throne...
2. How likley are the Serbs to continue resistance into the winter in the hope of eventual intervention?
It would be a bloodbath, the Serbs had been merrily killing Albanians and they would be out for blood, same for the Bulgarians, the Serbians might try but it would just make things bloodier.
3. Do Rumania, Bulgaria, or Greece haven any residual territorial or other issues that would bring their participation against Serbia?
Bulgaria is in head first, the 2nd Balkan war happened when Russia allowed Bulgaria to be gangbanged by all its neighbours, so expect Sofia to cut a deal with the Turks and join the trashing of the Serbs, expect some ethnic cleansing in Macedonia.
Serbia and Bulgaria defined their territories before attacking Turkey on the 1st Balkan war, after the war the Serbs refused to hand over the lands assigned to Bulgaria so the latter mobilized, the Russians were the guarantors but failed to pressure the Serbians to fulfill the treaty and even went to the Romanians and told them that they would not oppose them if they attacked whomever ended up being the aggressor in a hypothetical Balkan war... then they went to the Serbians and told them not to move first under any circumstances.
Then Romanians, Turks and Greeks joined in bashing the Bulgarian piñata (the Greeks signed a secret treaty with Serbia to screw the Bulgarians).
4. How bad would a Russian government or social crisis have to be before there was no chance of intervention by the following winter?
1906?
5. Are there any other nations that might intervene in a decisive way on the Serbs side?
Nobody, if the Russians dont move they are done. They had a pact with the Greeks but they would have to be foolish to jump in to get pasted along Serbia.