I am afraid I can't cover the entirety of your question, as it deals with different countries and their relations with the Nazi Regime. I also apologize for the large use of quotation marks but there is an inherent unpleasantness with the language, that I can't help but feel.

I have discussed the issue of Fascist antisemitism here but I'll summarize it a bit in the following. If you are looking more to the German relations with ther allies in the Context of the Holocaust, I hope someone else can provide further details.

As for Italy though, the Germans did pursue the extermination of the Italian Jews with remarkable vigor. Of the 7,495 Italian Jews collected by the Nazi occupation forces and the collaborating Salò Regime to be deported towards the extermination camps in the early months of 1944, 6,885 would die there.

To put these numbers in their right context it must be understood that the Fascist Regime as a whole never saw the extermination of the Jews as one of its central goals. There was no pre-existent "Jewish question" in Italy, since the Italian Jewish community was both small (around 50,000 - it would have probably been a bit larger if the German legisation had been adopted for census purposes) and well adjusted. While there is no conclusive statistical evidence, arguments could be made for Italian Jews being well represented both within the Regime and the State administration up until the turn of 1936-39 that led to the introduction of racial legislation in 1938.

While there is reason to believe that Mussolini - with other Fascist leaders such as Giuseppe Bottai - had in the late 1930s developed a more "racial oriented" vision of that conflict of peoples/races/cultures, that according to his view drove historical evolution, this still didn't bring him to advocate for any systematic extermination.

Racial legislation in Italy took the form of a sort of "segregation" as it was both modeled on the German Nuremberg Laws of 1935 (and the subsequent legislation) and inspired by the "necessities" of keeping the Italians from "mixing" with the locals in the new Italian Empire. Once this goal was accomplished the Italians did not consider surrendering their Jews to Germany. Such an action, besides offering no immediate advantage to the Regime, would also have risked antagonizing the Church - the Church looked with major concern at the presumption of a government to judge of the religion of its citizens.

Even in 1943, the first steps of the Salò Regime did not involve deporting the Jews (the Italian administrations applied here a sort of mild passive resistence - in short complying very slowly) but rather robbing them of whatever properties they had left to sustain the wavering finances of the puppet state.

To add to this those individuals within the Regime that were more inclined to cooperate with the Nazis - Roberto Farinacci and notorious antisemite Giovanni Preziosi - were not by the late 1930s on Mussolini's good side. Especially Preziosi was more or less openly disliked by him.

The result was that, until Italy remained for Germany a valuable ally, talks of surrendering the Jews were not openly held - I am not aware of general inquiries, but I assume they might have taken place to some degree after the war begun.

Only when the Fascist Regime fell and the Nazis - now in their fully radicalized state - propped up the puppet Regime of Salò, could they pressure Mussolini into establishing an office for race and demographics (March 1944) under Preziosi's control, which worked essentially as a "deportation office" that allowed the Germans to take those Jews that still lived in the regions controlled by the Salò State.

Those who lived in the South (or who had found some degree of protection in Rome) were thus able to escape.