So, first a note on terminology that I will use: The Nazis persecuted people they classed as "Zigeuner", a derogatory and racially charged term that is commonly translated as "gypsy", which while also used in a derogatory manner, has a bit of a different history of usage. The people classified as such by the Nazis were Roma, Sinti, and various other groups that because of their socio-cultural behavior were coded as "Zigeuner" by the society around them (e.g. Karner in the Tyrolean region were people who had no linguistic connection to Roma and Sinti but were also traveling people fulfilling the same functions and works in regional society as the Roma and Sinti elsewhere). I specifically use the terms "Zigeuner" and "gypsy" in quotes because a.) the terms are considered derogatory and members of the groups did certainly not identify as such and b.) therefore I am wary of reproducing the categories the perpetrators used to describe these people. Similarly to "asocial", it is advisable to not perpetuate these perpetrator categories in historical writing too much. It is still however useful to point out that who the Nazis defined as "Zigeuner" went beyond Roma and Sinti as distinct linguistic and cultural groups.

Secondly, Roma, Sinti and other groups still face discrimination, persecution, and are being ostracized in Europe to this day and have so since before the rise and persecution of the Third Reich. This has affected historical research into this subject in a variety of ways: The Porajmos, which is the name Roma give to the genocide exacted against them by the Nazis (similar to the term Shoah in a Jewish context), has only been officially recognized as such in Germany since 1982 for example. And because these groups have such a long and still ongoing history of discrimination and persecution, a lot of the transfer of knowledge of their own history is oral and thus was lost when so many of them perished and has further solidified a somewhat justified distrust towards outsiders, which makes researching the history of these communities during the rule of the Nazis and its impact even harder.

There are such big holes in our knowledge about the persecution of the group that to this day, there is considerable debate about the numbers of people persecuted as "Zigeuner" killed by the Nazis. While most historians estimate that 220,000 to 500,000 Romani were killed by the Nazis and their collaborators, or 25% to over 50% of the slightly fewer than 1 million Roma in Europe at the time, other scholars dispute the number of 1 million Roma in Europe all together (seeing it higher) and arriving at estimates of 1.5 million people killed.

The questions you ask here could (and have) filled whole books by themselves but I will try to give a short overview/introduction of what we know:

So, the best researched area in connection to this is certainly the treatment of Roma and Sinti and similar groups in Germany itself. While the discrimination of Roma and Sinti and others certainly has a long history in Europe overall because of their status as epitomizing the "other" in a certain sense, the 19th century and the rise of völkisch race theories saw a lot of discrimination based on this come to a head. During this time of the rise of nationalism, Social Darwinism, and scientifism the concept of race was systematically employed in order to explain social phenomena. At the same time, modernization lead to not only greater control of the state over its subjects but specific campaigns to discipline people in accordance with the new modern system, which included personal papers, the furthered reach of bureaucracy and the push to make them settle down and take part in the new capitalist mode of economics.

All this hist Roma, Sinti, and similar groups hard for their life-style that did not fit with the new modern modes of production paired with old prejudice lead to them being deemed an "inferior race" of peoples, naturally prone to criminality, and other socially deviant behavior. Even in the Kaiserreich as well as in Weimar several laws were enacted to either make them settle down and force them into taking part in capitalist modes of production or to restrict their travel as a whole. And this is where the base for the later prosecution and murder of the Nazis comes into play: The Nazis viewed people they deemed as "Zigeuner" as raciall inferior and racially prone to crime and socially deviant behavior. They had already stigmatized as habitual criminals, social misfits, and vagabonds and the Nazis, in typical fashion, radicalized their policy of "fighting against" these manifestations.

While the beginning of Nazi rule saw a massive expansion of the police mandate to discriminate against Roma, Sinti, and others, including sending them to Concentration Camps as "asocial", no systematic policy had been formulated. To that end, the SS hired "racial scientist" Robert Ritter to conduct a through "racial" examination of the Reich's "Zigeuner" population. And so, Ritter set out to do just that with the people "examined" in the context of his study often facing further discrimination and imprisonment.

As a result of the study Heinrich Himmler suggested a comprehensive "Zigeuner law" that should consiste of the following:

The aim of measures taken by the State to defend the homogeneity of the German nation must be the physical separation of Gypsydom from the German nation, the prevention of miscegenation, and finally, the regulation of the way of life of pure and part-Gypsies. The necessary legal foundation can only be created through a Gypsy Law, which prevents further intermingling of blood, and which regulates all the most pressing questions which go together with the existences of Gypsies in the living space of the German nation.

In essence, the idea was to deport them to a remote reservation in order to create a spatial barrier towards them (as with "territorial solutions to the Jewish question", these plans already carried the seed of genocide in them as in the course of these deportations and in these reservation, there was no doubt that many of them would perish).

And while this law never came, discriminatory measure were still implemented. People deemed "Zigeuner" were from the mid-1930s forward placed in special camps and when ghettos for Jews were set up and the deportation of Jews started during the war, "Zigeuner" were included in these measures, including being shot in the Soviet Union or being killed in the Reinhard Death Camps as well as in Auschwitz, where there even was a separate "Zigeuner" camp within the Auschwitz camp structure.

With the persecution and murder of Jews and Roma, Sinti, as the only groups systematically murdered during the Holocaust, you can really see two seperate manifestations of racism as a phenomenon: While in the case of anti-Semitism, the basis was an alleged devious conspiracy of Jews as an alleged race adept at subverting nations and destroying them from within, anti-Ziganism saw Roma, Sinti, and others as dangerous because of their primitivity, proneness to crime and social deviance of a different sort of the Jews. The latter mirrors "classical" racism much closer than anti-Semitism but was still not less potent.

And while still in the process of being researched, the development of anti-Semitic murder policy and anti-Ziganist murder policy does mirror each other: For it was the attack on the Soviet Union when the Einsatzgruppen started murdering Roma, Sinti, and others together with the Jews in the summer of 1941 in a systematic fashion. And like with the Jews, the, let's call them, nuances of the policy within the Reich were quickly forgotten when it came to the murder in the occupied territories.

Like with Jewish Mischlinge in Germany, the Nazis originally distinguished in accordance with Ritter's study between "pure blood Zigeuner" (mostly Sinti), who were not slated for immediate killing but for reservations and "mixed blood Zigeuner" (mostly Roma) the ones to be killed. But as with ideas of Mischlinge, in the Soviet Union this went out of the window pretty quick because the Einsatzgruppen relying on local knowledge about who was Jewish and who was "Zigeuner" started killing people they just suspected as belonging to these categories. A similar undifferentiated approach in the Balkans would also cause massive tensions in the area because many Roma in the area were Muslims and thus felt under the protection of the satellite regime in Croatia, a fact for which the Nazis cared little.

While there has been more extensive research on local phenomena of persecution and killing with German Roma, Sinti, andothers through study of such camps like Lackenbach, a transit camp in Austria from which many were deported to Auschwitz and Ghettos in the East, the local dynamics of persecution and murder in the Soviet Union and elsewhere are still being research, mostly by a colleague of mine, Martin Holler, who has been working for more than 10 years on the topic.

It also bears mention that in terms of relations to the majority population in their occupied countries, I have written before about Roma in Yugoslavia and especially with the socialist Partisans there, relations were generally positive and where the Tito Partisans in line with their general tactic of the struggle for national liberation promised the Roma of Macedonia – which were more numerous and identified more directly as Roma than their Bosnian compatriots – that they would establish an autonomous Roma region in Macedonia (akin to the Kosovo autonomous region in Yugoslavia after the war) once national liberation was achieved. The plan ultimately fell through because of Bulgarian resistance to it but it is an interesting example nonetheless.