At that point Italian territory has fall to British forces so not much reason to give Ethiopia anything. Rather give up their own territory, would be more likely to just reverse Italian gains of Hoare-Laval Pact. Without the need to compensate Ethiopia for the Italian occupation Eritrea might go to Britain.
Maybe. Again, it depends on what Italy does in this situation.
I get the vibe that Italy is more likely to be Neutral or Allied than Axis in this situation. Ethiopia had 800,000 soldiers during the Italo-Ethiopian War, so an Ethiopian offer to contribute troops wouldn't be that implausible.
Although I suppose there's a different situation that we aren't even looking at: Ethiopia could join the Axis. OTL Germany sent three planes, 10,000 rifles, and 10million rounds of ammunition to Ethiopia during the war. If Hoare-Laval goes through and Stresa holds up, we might see some quirky outcomes. France and Britain have betrayed Ethiopia. France and Italy were for a tough line on Germany whereas Britain was trying to play both sides. I can see a situation in which the French and Italians are willing to get into a fight with Germany over Austria (or perhaps even the Rhineland), but the British are unwilling.
@Jackson Lennock, do you think that the Ethiopian government would be a Fascist one in this scenario?
I don't know enough about Ethiopia's internal politics at the time. I think if Britain and France are sticking with Italy, it wouldn't be wild for Ethiopia to look to Germany and Japan for aid. Germany supported Ethiopia a bit OTL just to spite Mussolini and the Japanese and Ethiopians had some interesting long-standing relations. That doesn't necessarily mean the country would be Fascist though.
If German advisers and weapons started popping up in Ethiopia in 1936, I wonder how impactful that would be in a war that starts in 1938 with Germany (and Ethiopia) on one side and Italy-France-Austria on the other.