It depends on how you define European and from whose side are we looking. The Ottomans themselves come from the House of Osman, a noble house of the Sultanate of Rum. There was an immense amount of inter-marriage between them and other Turkic dynasties and the Byzantine ruling families and their power base in Anatolia was the same as the old Byzantine empire. In fact, the idea of what we call Turkey now as non-European is a very recent one, as Anatolia was a foundation of the Greek world for a very long time and up until quite recently had an immense Greek population.

While not considered a part of Christendom, for those who used it as the primary definition of European, the Ottoman empire was seen as European by many others as it was primarily a European empire. It's most important territories were in the Balkans and Anatolia, much of the ruling class came from European (Balkans) or Europeanized (House of Osman, Turkish coastal regions) areas of the empire and much of its geographic focus was on Europe, with Persia usually playing a more minor role depending on how hot the competition was for Iraq at the time.

Sources: Fahmy All the Pasha's Men, Kafadar, A Rome of One's Own

Absolutely not. The Ottomans were always viewed as the scourge of Europe and a threat to the European way of life. The nations of Europe did a very good job of banding together and creating a cultural identity known as "Christendom" that took root during the crusades, you could argue this mentality is what saved further Islamic expansion before the times of the Ottomans and what ultimately stopped their advances into Europe. For example, after the Battle of Lepanto, which was a defining naval battle that meant the end for further Ottoman expansion along the Mediterranean, there was widespread celebrations across the whole of Europe even in countries that weren't involved in the battle and despite a lot of Europeans being at odds with each other in the Catholic/Protestant conflicts. After the Battle of Vienna, which marked the end of Ottoman conquests in Eastern Europe, the Polish ruler John Sobieski was dubbed the saviour of Christendom. Despite their differences everyone in Europe was still very unified in their hate of the Ottoman Empire and they still viewed it with an us vs them mentality and viewed the Turks as foreign invaders of Europe.

Now, moving on to the phrase "sick man of Europe". It's important to note that when Tsar Nicholas I of Russia actually spoke about the Ottomans in this way, he never actually said "of Europe". When speaking about the Ottomans he merely said "a sick man—a very sick man," a "man" who "has fallen into a state of decrepitude", or a "sick man ... gravely ill". The first time the term "sick man of Europe" was actually seen was in the New York Times when a journalist was referring to the Habsburg Monarchy, an actual European state.