This answer from u/justcoffeeok might be helpful and sums up a lot of what I am going to say. Unfortunately I cannot provide a satisfactory answer to all of your questions, but I will of course answer as best as I can.

Wilhelm's attitude towards the Hitler and the Nazis is complicated and influenced by numerous factors, primarily the restoration of the monarchy, the relationship between the old and new right, and German nationalism. Members of the ex-royal family took a liking to the Nazis, especially his second wife Hermine, who attempted to set up a personal meeting between Hitler and Wilhelm, and his son August-Wilhelm, who was a member of the SA. While Hitler and Wilhelm never met, Herman Göring made multiple visits to Wilhelm's residence at Huis Doorn where he assured the ex-kaiser of the Nazi's commitment to restoring the monarchy. Furthermore, Mussolini's relationship with King Victor Emmanuel III of Italy gave Wilhelm further reason believe Hitler would do the same.

This did not translate into fanatical support though and the ex-Kaiser took issue with Hitler's low-born status writing to August von Mackensen, "The new Reich will not come from a beer joint," as quoted be Lamar Cecil in biography of Wilhelm.

Wilhelm's attitudes toward Hitler began to sour in 1932 when Hitler refused to become vice-chancellor under Franz von Papen. This did not necessarily impact his attitude towards the Nazi Party though, believing that "The strong National forces that are even today still present in the Nazi-Party must be rescued from the totally irresponsible demagogic hustle and bustle of some of the leaders and orators and brought behind the national government," as written in a letter to the Crown Prince and quoted in John Röhl's Wilhelm II: Into the Abyss of War and Exile. Wilhelm's dreams of a restored monarchy came crashing down in 1933 when the Nazi's began consolidating power. The Night of the Long Knives in 1934 for example "reminded Wilhelm of Chicago gangsterism," according to Cecil and left him with little hope as most the the remaining elements of the establishment were purged.

Much like how Wilhelm could separate his hatred for Hitler from the rest of the Nazi party, before events like the Kristallnacht soured his attitude towards the latter, he could also sepearte his hate for the Third Reich from his German nationalism. The ex-Kaiser was excited to greet German soldiers on the steps of Huis Doorn when they arrived in 1940. He furthermore sent his congratulation to Hitler after the fall of Paris stating, “Under the deeply moving impression of France’s capitulation I congratulate you and all the German armed forces on the God-given prodigious victory with the words of Kaiser Wilhelm the Great of the year 1870: ‘What a turn of events through God’s dispensation!’” as quoted by Röhl. Even this though was not enough to restore restore relations with the Third Reich and so the ex-Kaiser requested that no Nazis be present for his funeral, which the latter of course did not respect because of the propaganda value it had much like earlier associations with the ex-royal family during their rise to power.

Here are the three biographies I used when I wrote a paper about this a few years ago. I was only a sophomore so my apologizes if the scholarship is lacking a bit.

Cecil, Lamar. Wilhelm II. Vol. 2. Chapel Hill: University of North Carolina Press, 1996.

MacDonogh, Giles. The Last Kaiser William the Impetous. London: Weidenfeld & Nicolson, 2000.

Röhl, John. Wilhelm II: Into the Abyss of War and Exile, 1900-1941. Translated by Sheila de Bellaigue and Roy Bridge. Cambridge: Cambridge University Press, 2014.

TL;DR: the Nazis didn't like him, but used his dynasty to improve their party's standing, the people were still affectionate and many remained ardent supporters.

What was the attitude of the Nazi party leadership to the exiled emperor?

Hitler had stated in Mein Kampf, which Wilhelm never read, that National Socialism’s goal was not ‘the foundation of a monarchy’ rather ‘the creation of a Germanic state.’ The Kaiser’s youngest son, August Wilhelm (Auwi), had joined the SA in 1928 and he was used by Hitler to acquire the votes of many who still supported the monarchy. However, Hitler had no interest in restoring the monarchy and took actions to prevent monarchist parties and groups from forming. On the 27th June 1934 Hitler ordered celebrations of the Kaiser’s birthday to be broken up and swiftly passed legislation outlawing monarchist organisations, the army took vows to Hitler that year while in 1935 the Imperial black-white-red was banned. Hitler would later refer to the Kaiser as having been a ‘strutting puppet of no character’.

Goering (as the aristocratic face of the party) met with the Kaiser in Doorn in January 1931, however no minutes were kept of the meeting. After meeting Goering again in 1932 Wilhelm believed his return to the throne was certain with Goering’s support in Berlin, however, when Goering heard news of the Kaiser stating that Hitler ‘is no statesman’ and threatened to tell the Fuhrer, the Kaiser was outraged, and Goering was never invited back to Doorn, the Kaiser having recognised that Goering’s proclaimed intentions were not genuine.

The Kaiser only ever met Goering in person and was in contact occasionally with Hitler. Goebbels, never met the Kaiser but described him as an ‘‘incorrigible fool’ who was probably of part-Jewish ancestry.’

The Kaiser had seemingly realised the Nazi intentions were not to reinstate the monarchy by the mid 30s and so he effectively cut ties immediately, spending the remaining antebellum years complaining to anyone who would listen about the Nazis' actions reminding him of 'Chicago Gangsterism'.

General Public’s view?

The public’s view was generally not particularly favourable towards the Kaiser, blaming him for losing the war, believing memoirs published by generals desperate to clear their name. However, the Kaiser retained many supporters who would send birthday wishes and such and on his 70th birthday in 1929 he gathered many of his old generals. The greatest evidence of the public’s affection for the monarchy is evident in how they reacted to royal deaths. When the hugely popular wife of the Kaiser, Augusta Victoria died in 1922 mourners lined the road as her casket passed, reportedly forming an ‘unbroken human chain’ and in Berlin it was reported that at least 200,000 mourners had gathered. A similar situation became apparent in 1940 when the Crown Prince’s eldest son, Wilhelm was killed in action fighting in Northern France. At least 50,000 mourners turned out for the Prince’s funeral and this deal of support shocked Hitler so greatly, that he banned all members of royal houses from serving in the armed forces soon afterwards, hoping to prevent more royal martyrs being made.

Overall, the Nazi officials certainly had no love for him, but the people remained rather affectionate of their royal family and many hoped for an eventual restoration of the Hohenzollern dynasty. The army remained strongly monarchist, with most officers still coming from aristocratic families, but this meant little in terms of their actions before and during the war. Although I have read stories of German soldiers coming to meet the old Kaiser in great numbers, before Hitler placed the Kaiser under permanent SS guard.

Sources:

Clark, C. (2013) Kaiser Wilhelm II, Abingdon and New York: Routledge.

Cecil, L. (1993) Wilhelm II Volume 2: Emperor and Exile, 1900-1941. Chapel Hill: The University of North Carolina Press.

Röhl, J. (2014) Kaiser Wilhelm II, 1859-1941, A Concise Life. Cambridge: Cambridge University Press.

This question comes up quite a lot, but detailed replies seem to be rare (or my search-fu is lacking).

Still, I found two answers in different threads, that might help answer parts of your question.

What happened to Wilhelm II by u/loudshell

What happened to Wilhelm II after Nazi Germany invaded The Netherlands? by u/RicardoHuch