This is a great question, /u/EnclavedMicrostate! The involvement of the Ottoman Empire in the Thirty Years' War is an often overlooked, but interesting, aspect of the conflict!

Introduction

To start with, the Ottoman Empire never officially entered the conflict and their involvement in the overall fighting was quite limited. With that being said, the Ottomans were still partially involved in the conflict through supporting their vassal, Transylvania, which partook in the fighting. And the Ottomans did host an international array of Austrian, Spanish, French, Dutch, Swedish, and other foreign diplomats, who all worked to advocate for their side and influence the sultan. Therefore, this answer will detail the Ottomans role—and lack of role—in the conflict and how they interacted with European states involved in this war.

I'll try to answer your questions one-by-one, but let me know if you'd like any more information on a point.

"What was the Ottoman Empire up to in Europe during the Thirty Years' War?"

From 1618 to 1648, the Ottomans were actually involved in a series of European wars, but, while these were often against states involved in the Thirty Years' War, these conflicts were rather periphery to the overall European war.

Early in this period, the Ottomans found themselves at odds with Spain (1618–1619) and Poland (1620-1621), but both of these wars were rather quick and inconclusive. Some Ottoman historians, such as Mahmut Halef Cevrioğlu, have argued that these wars should be considered part of the Thirty Years' War and were in indirect aid of the Anti-Imperialist powers.

This may be possible, as we do know that the Ottomans were interested in interfering in the Bohemian Revolt (1618–1620). First, the Ottomans, wanting to further stabilize the region, sought to supply troops to Austria, which were declined.^(1) However, upon seeing the early successes of the Revolt, the Ottomans entered negotiations for an alliance with the rebels in 1619.^(2) While this looked promising, the alliance fell through when the Bohemian Revolt's fortunes turned and quickly collapsed in 1620.

But it's actually not what the Ottomans were directly involved with that is interesting during the Thirty Years' War, as these early conflicts on spanned a few years. Instead, their vassal, Transylvania, is actually the big story here. Early in the conflict, the Ottoman sultan granted Transylvania permission to the enter the Thirty Years' War. Soon Transylvania found much success in invading Hungary and did so almost yearly from 1618 to 1626.

These early success in Hungary soon earned the Transylvanian prince a visit from French, British, and Dutch diplomats, who were eager to strike an alliance with the princedom. Ultimately, permission had to be granted by the Ottoman sultan in Istanbul, who agreed on several occasions to these alliances spanning to the end of the Thirty Years' War.

While the Ottomans weren't too keen on being directly involved in these offensives, they did augment the Transylvanian army at times with small forces. However, in 1626, when the Austrian diplomat in Istanbul complained that supplying the Transylvanian army with Ottoman forces violated the Austria-Ottoman Peace of Zsitvatorok, the Ottomans quickly withdrew their personal forces.

To the end of the war in 1648, Transylvania would continue to find success on the battlefield and worked closely with the Anti-Imperial powers to achieve the Peace of Westphalia.

In addition, sometime around 1630, Sweden also struck an alliance with the Crimean Tatars, which were another vassal of the Ottomans. But this fell through when the Crimean Tatars were diverted by the Ottoman sultan to the Safavid front during the Ottoman-Safavid war (1623–1639).

From this discussion, we can see that the Ottomans never fully committed to the Thirty Years' War. It was mainly their vassals, Transylvania and to a lesser extent Crimea, with their permission and support, that played a role in the conflict. Some of the Ottoman wars during this period have been considered to be a part of the larger European war, such as the Ottoman-Spanish and Ottoman-Polish wars, but this area isn't too well studied or agreed upon.