There were certainly plenty of Ukrainians who collaborated with the Nazis and actively participated in the Holocaust, but to portray it as unique or to suggest that all (or even most) Ukrainians willfully collaborated would be ahistorical and dishonest. The phenomena of collaboration and resistance existed throughout Nazi occupied Europe, and people had a variety of reasons for collaborating: ideological alignment with the Nazis, opposition to the previous government (particularly in the occupied Soviet Union), economic gain, or simple survival. All of these were present in occupied Ukraine, as they were elsewhere in Europe.

To answer your first question, the relationship between Ukrainian nationalists and the Nazis was complex and fractious. Ukraine itself was not a politically cohesive entity, with the western parts of the Ukrainian SSR (which had been part of Poland prior to the Soviet invasion in 1939) being generally more nationalist and anti-Soviet than eastern Ukraine. The most prominent Ukrainian nationalist organization was the Organization of Ukrainian Nationalists, which was split into a more moderate faction (OUN-M) and the more radical (and more famous) OUN-B, led by Stepan Bandera. The OUN-B had political leanings that could be described as fascist or quasi-fascist and were strongly opposed to Soviet rule, and thus welcomed the Germans as liberators who freed them from Russian rule. They saw the German invasion as an opportunity to establish an independent Ukrainian state, which the Germans obviously had no intention of allowing. Despite the fact that the Germans and Ukrainian nationalists shared a common enemy, the Germans still viewed the Ukrainians as Untermenschen, like other Slavs, and eventually repressed the OUN; most of occupied Ukraine was eventually incorporated into Reichskommissariat Ukraine, one of the five planned administrative divisions of the occupied Soviet Union. The OUN and other Ukrainian nationalist groups later participated in violence against ethnic Poles in the western part of the present-day Ukraine (Volhynia and Galicia) in 1943-1944. There was also day-to-day collaboration on the part of normal Ukrainians in the form of economic relations with the German occupiers, which, while framed as collaboration after the war was frequently simply a way for them to survive under wartime conditions. This kind of morally ambiguous cooperation with an occupying force was common throughout occupied Europe, as were postwar controversies and recriminations related to it.

To answer your second question, there were a number of Ukrainians who participated in the Holocaust, both in the "Holocaust by bullets" in the East and in the Nazi concentration and extermination camps. In the former case, Ukrainian auxiliary policemen and Ukrainian SS volunteers (which included a large number of members of Ukrainian nationalist organizations like the OUN) participated in a few different roles, assisting in rounding up Jews and transporting them to the execution sites, guarding the execution sites, and, in some cases, pulling the trigger themselves. The Nazis frequently used local auxiliaries to augment their forces during the killing operations to allow them to operate more efficiently with fewer German personnel committed; there were similar instances of collaboration elsewhere in the east, most notably in Lithuania and Latvia, where the local Jewish communities were almost completely destroyed (>90% killed) thanks in large part to extensive local collaboration. There were also pogroms and instances of looting of Jewish property in Ukraine in the wake of the German occupation, as there were in other parts of the German and Romanian-occupied Soviet Union.

In the latter case, several thousand Ukrainians were trained (principally at the Trawniki camp) to serve as guards in the Nazi concentration and extermination camp system. These men were former Red Army prisoners of war who were released from the POW camps in exchange for their service at the camps. I've written about this phenomenon previously, but essentially, due to the horrific conditions and extremely high death rates in the German camps for Soviet POWs, the choice for many of these men was "collaborate or starve". You can decide for yourself how you feel about the moral implications of their decision, but it's not surprising that thousands of men who faced the prospect of starving to death chose the other option. Some of these guards who survived the war were later prosecuted for their participation in the Holocaust, the most famous case being that of John Demjanjuk, who was deported from the United States to Israel and tried for serving as a guard at Treblinka, convicted under controversial circumstances, and then had his conviction overturned. Again, these instances of Ukrainian collaboration have a complex historical and present-day legacy that doesn't lend itself to simplistic statements about Ukrainians as a whole.

To answer your last question, no, collaboration in Ukraine wasn't unique, as the previous answers have hopefully demonstrated. The question of collaboration with Nazis and participation in the Holocaust was and has remained highly controversial in most countries that were part of Nazi-occupied or Nazi-allied Europe. Most countries had fascist and nationalist groups that aligned themselves with the Nazis, much like the OUN-B in Ukraine (for example, the Rexists in Wallonia, the Nasjonal Samling in Norway, and the Perkonkrusts in Latvia), and many of these groups actively participated in Nazi crimes. Direct participation in the Holocaust and killing of Jews and other civilians was more common in Eastern Europe because most of the killing on the Eastern Front took the form of mass shootings rather than deportation to the extermination camps, as was the case in Germany, Western Europe, and Poland, but even then, as I noted above, local auxiliaries and SS volunteers participated in these crimes throughout the occupied Soviet Union, not just in Ukraine. In that sense, collaboration in Eastern Europe is different from collaboration in Western and Central Europe, but that characterization applies basically across the board in Eastern Europe. On the whole, Ukrainians weren't significantly more enthusiastic about collaborating with Germany than other countries, but there were some unique circumstances in Ukraine that gave Ukrainian collaborators a more prominent, visible role than other groups of collaborators whose actions were just as violent, but not as well-known to most people who aren't intimately familiar with the history. Without getting into present-day issues and running afoul of the twenty years rule, I'll just say generally that to present Ukrainian collaboration as unique for propaganda purposes is just that, propaganda, and should be regarded with the same skepticism and subjected to the same rational inquiry as any other politically-motivated claims about history should be.

Sources:

There are two very good recent books on the subject that I'd recommend if you're interested in the question of collaboration in Ukraine and in occupied Europe more generally: Martina Bitunjac and Julius B. Schoeps, Complicated Complicity: European Collaboration with Nazi Germany during World War II (De Gruyter, 2021) and Paweł Markiewicz, Unlikely Allies: Nazi German and Ukrainian Nationalist Collaboration in the General Government during World War II (Purdue UP, 2021). There are some additional sources which deal with the participation of Ukrainians in the Holocaust (and particularly the Trawniki men) in the previous post of mine that I linked above as well.