The historical record doesn’t match your claims on two of the three points, and is more nuanced on the third. 
Guns. The Weimar Republic already had strict firearms controls (registration and permits under the 1928 law, following even tighter post-WWI measures). The Nazis’ 1938 German Weapons Act relaxed rules for long guns, ammunition, and many “reliable” Germans—including Nazi Party members, hunters, and government workers—while specifically barring Jews from the firearms trade and, after Kristallnacht, from possessing any weapons at all. They used existing registration records to confiscate guns from political opponents early and from Jews later. Himmler ordered concentration camp sentences for Jewish firearm possession. This was selective disarmament of enemies, not a general push for civilian gun control in the modern sense. Loyal Germans got easier access; targeted groups did not. Claims that Nazis broadly “promoted gun control” as an ideology are overstated; claims they were simply pro-gun for everyone are also false. 
Abortion. Nazis did not promote abortion as a general policy or right. For “Aryan” Germans they restricted it, suppressed the birth control movement, and imposed severe penalties (including, later, potential death penalty for German women). They pushed large families and population growth for the desired race. Eugenic exceptions and forced measures applied to the disabled, the “hereditarily unfit,” Jews, Slavs, and others deemed undesirable—sterilizations under the 1933 Law for the Prevention of Offspring with Hereditary Diseases, and abortions on racial grounds especially in occupied territories to reduce non-German births. The policy was racial hygiene and demographic engineering, not liberal abortion access. Your point holds on the core claim. 
Government. Nazis ran a totalitarian dictatorship under the Führerprinzip: Hitler’s will superseded written law, the Party fused with the state, opposition was eliminated, and loyalty oaths were to Hitler personally. Propaganda, the Enabling Act, Gleichschaltung of institutions, control of media/education/economy (war economy), and the cult of the Führer made the regime the center of authority and identity. “Ein Volk, ein Reich, ein Führer” was the operating slogan. Private property and markets existed but were subordinated to state goals. This is textbook authoritarian state worship via the leader and party, not limited government or anti-statism. The claim that they “certainly didn’t worship the government” is not supported by the structure or practice of the regime. 
Modern neo-Nazis. They vary. Many in the US and elsewhere are strongly pro-gun (often explicitly to arm against the state or for conflict), anti-current-government (viewing liberal democracies, immigration policies, and institutions as controlled by enemies), and socially traditionalist on family/reproduction. They typically romanticize Nazi-era authoritarianism, racial hierarchy, and ethnostate goals while rejecting the existing state. Overlap with historical Nazis exists on racial ideology and anti-liberalism; differences are large on the practical relationship to the government of the day. “Don’t differ a whole lot” flattens both the historical totalitarianism and the contemporary anti-state posture.
The actual record is selective disarmament plus racial eugenics plus extreme authoritarian centralization. Simplified political talking points on either side tend to omit the selectivity.