This is a very complex question that has lead to decades of controversy. As such I can only give you a relatively short rundown. Beware: I'm German so excuse my English. If you don't understand stuff, just ask.

Initially Heidegger was a strong supporter of National socialism which is shown in his private letter conversations. He admitted to reading the Nazi newspaper Der Stürmer in a letter in 1930 to his wife. He admired the second half of Mein Kampf, which depicts the ideological foundation of Nazism in a letter he wrote to his brother in 1931.

Additionally he recommended his brother reading far right and antidemocratic sources like Deutschland in Ketten.

He voted NSDAP in 1932. He became a party member in 1933 and stayed in it until the end of the second World war.

His most open and strongest ideological admiration was the Führerprinzip. He whorshipped Hitler as the only one who could save the Occident. Heidegger looked upon himself as the philosophical counterpart to Hitlers conservative revolution. He held a plethora of speeches before his students praising Hitler and signed an endorsement of German professors towards the Führer in 1934.

He even went as far as restructuring his university after the Führerprinzip and establishing an upper limit of "non-aryan" students at his university (1,5%).

Heidegger imself was antisemitic. Discussion about wether or not this influenced his philosophy has started with the release of his Diaries, the Black Notebooks, in 2014. I'm not qualified enough to answer this qustions but it is clear that he was anti-semitic. Despite that he actually helped jewish students at his university.

Heideggers opposition to Nazism was mainly due to his wish for a continued conservative revolution which the Nazis couldn't offer. As early as June 1933 Hitler said that revolution couldn't be a permanent condition for Germany.

This was contrary to what Heidegger wanted. He projected his own anti-modernism and antisemitism into the national socialist movement. He wanted it to be a continued, metaphysical and philosphical reactionary backlash against the progress he feared: a "Heideggerian phantasy fascism" (Richard Polt: Heidegger: An Introduction, 1999).

The Night of the long Knives was the clear statement against a continuation of the Volksrevolution in form of the revolutionary wing of the NSDAP, the SA. As such Voctor Farias argues that this event was the turning point for Heideggers relationship to Nazism (Heidegger und der Nationalsozialismus, 1989). he supports this by the fact that he stopped his political activities for the NSDAp in the same year.

The last part of the question is unfortunately too much too handle. If you're interested, there's a a plethora of writers, historians and intellectuals who have discussed this topic over the last decades, defending or criticizing him. This has come in waves with new approaches or eveidence coming up. Relatively recent examples are Nolte, Farias and Faye.

Sources about his ideology (German):
http://www.zeit.de/2015/11/martin-heidegger-nationalsozialismus-schwarze-hefte
http://www.deutschlandradiokultur.de/umstrittender-philosoph-ueberlegungen-zum-judentum-die.954.de.html?dram:article_id=272402

I will try to give you a very short answer to your bonus question. Keep in mind im writing this in the perspective of critical theory, which is imo most capable of explaining the connection between nazism and heideggers philosophy -and ultimatively its roots in german romanticism. First you have to understand what the ns-ideology and its core, antisemitism, was. Antisemitism is very different from racism, its an ideology which imagines a global jewish conspiracy. Rationality, the modern and the abstract are associated with the jewish. Hence phenomenons of the modern like communism, global capitalism/globalisation , democracy, cosmopolitism, the mass media and rational science were labeled "jewish". On the contrary the nazis positively referred to Hard (physical)work, naturality and mysticism/the irrational volkisch as germanic. Heidegger unites admiration of nature, mysticism (metaphysics) and the rejection of modernity/enlightenment in his postmodern philosophy. The connection between heidegger and nazism should be Pretty clear. For further reading and source: poistone, moshe: antisemitism and national socialism.