Because that is the consensus of reliable sources, in this case historians and political scientists. Almost all historical and present-day academic literature places the Nazi Party on the far-right of the traditional left-right spectrum, which in turn is the most common short-form classification used in political science. The Nazis themselves attacked both left-wing and traditional right-wing politicians and movements in Germany as being traitors to Germany. While the Nazi regime's economic policies are very different from those of present-day right-wing parties that adhere to classical liberal or neoliberal positions (which advocate, e.g., a highly deregulated, privatized economic environment), Nazi economic policy was typical of the early to mid-twentieth century far-right, and indeed most political currents of the time, in that it embraced interventionist economics. The Nazi Party absorbed the far-right reactionary monarchist and nationalist German National People's Party into its membership in 1933. The Nazi Party also held good relations with openly right-wing political movements in Europe, such as the Spanish Confederation of the Autonomous Right, whose leader Gil-Robles was a guest at the 1933 Nazi Party Nuremberg rally and sought to model his movement upon the Nazi Party.

Many political entities have names that can be misleading. Consider, for example, the Holy Roman Empire (a confederation of mainly German territories during the Middle Ages and the early modern period) and North Korea's official name, the Democratic People's Republic of Korea (a totalitarian dictatorship). Historically, several right-wing figures used the term "socialism" to mean something very different from what would be understood by traditional left-wing socialism, referring simply to the broader concept of collectivism and anti-individualism. The prominent French reactionary monarchist Charles Maurras famously said "a socialism liberated from the democratic and cosmopolitan element fits nationalism well as a well-made glove fits a beautiful hand". Maurras's views influenced fascism. Oswald Spengler's ideal of "Prussian Socialism" directly influenced Nazism, and Spengler promoted it as a member of the far-right Conservative Revolutionary movement. The usage of the word "socialism" by the Nazis is different from the common usage of the term "socialism", which refers to an economic philosophy involving advocacy for social ownership of the means of production.
In the case of the Nazi party, the phrase "national socialist" was a nationalist response to the rise of socialism in Europe by offering a redefinition of "socialism" to refer to the promotion of the interests of the nation, as opposed to ideas of individual self-interest. But there was no policy of social ownership of the means of production. The Nazis did talk about capitalism being bad, but they defined it as a Jewish-originated economic philosophy based on individualism that promoted plutocracy in the interest of the Jews, at the expense of non-Jewish nations and races. This was put in contrast to the Nazis' conception of socialism, which was done in order to win over people attracted to anti-capitalist and socialist ideas to their cause. They rejected ideas of equality and working class solidarity, instead advocating for social hierarchy and national strength. This article sums it up well.

The answer depends on the context and definition of capitalism. Ideologically, Hitler in private was just as opposed to the ethos of capitalism as he was in public as a politician; he regarded the capitalist ethos as being self-centred individualism that was incompatible with nationalism. Furthermore, in both public and private Hitler regarded capitalism as being created by the Jews for their own interests. The Nazis in public and in private held contempt for bourgeois culture in liberal capitalist societies - as they associated such bourgeois culture with a cosmopolitan, liberal, and decadent lifestyle that was incompatible with the Nazis' ideal of a nationalist martial ethic of disciplined soldiers who were collectively committed to the Fatherland above any individual interest. So, ideologically, Nazism held strong antipathy to capitalism. However, at the same time, Hitler and the Nazis endorsed private property and private enterprise and did not challenge the market economy, which was important to their accrual of power because it avoided antagonizing industrialists and aristocrats. The Nazis themselves claimed that "true socialism" did not involve the Marxian opposition to private property. But if capitalism is defined in a minimum way as involving the support of the existence of private property, private enterprise, and a market economy, then from that minimum definition, the Nazis could be considered as endorsing a capitalist economy.

Because English Wikipedia uses the common name for things, and the common name for the NSDAP in English is the "Nazi Party".

Nothing. See this discussion where the community came to a consensus that we have entertained the numerous questions and claims about the Nazis being left-wing enough, and that continued engagement with people pushing this line of reasoning is not helpful to the article.

See the following links, all of which are to discussions about this very question over time. Any argument someone thinks is novel has already been made, been responded to, and failed to convince anyone. [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28]

Then you will be more than welcome to show them to us, so that we can see that they are very reliable and that they assert that Nazism is a left-wing ideology. If they are, then we will change the article.