What was the status of Shia Muslims in the Ottoman Empire?

Short answers first:

Heavily depended on circumstances, but on average rather poorly;
No;
Yes.

Technically speaking, there wasn't a divide between Sunni and Shiʿi millets—both fell under the same "Muslim" category. While this might seem to guarantee equal treatment of both sects, however, the Ottomans tended not to recognize Shiʿism as a valid form of Islam, and practitioners came in for a wide range of persecutions. These ranged from deportation and straight-up execution (during the Ottomans' wars with Safavid Iran, on which I've written a bit here) to "softer" forms like a ban on marriage between Sunnis and Shiʿites (officially mandated in 1822, but a subject of discussion among Ottoman clerics since the sixteenth century). Nor, unlike Ottoman Catholics or Jews, did Shiʿites have access to special courts for judging cases between co-religionists—though in practice they often took such cases to their local religious authorities anyways, rather than the official (Sunni) Islamic courts.

There was, however, a brief period in which this all seemed like it might change.

As I mentioned, one of the major drivers of early Ottoman anti-Shiʿism was the rivalry between the Ottoman and Safavid empires, which expressed itself in intermittent wars from around 1514-1639 and peacetime politicking until the near-total collapse of the latter state in the 1720s. So, in total, around 200 years of geopolitics, enough to form some very comfortable habits about the status of Shiʿism and how its practitioners should be treated! In 1736, however, a new ruler of Iran was enthroned, one who had rather novel ideas about the relationship between the two sects. In a speech to mark the occasion, Nader Shah indicated that while he planned to maintain Iran's official Shiʿi status, there would be some departures from Safavid traditions—in particular, that inflammatory practices such as the denial and public vilification of the first three Caliphs (a hallmark of Safavid Friday sermons, and a major cause for Ottoman complaint) would be discontinued in favor of a policy of accommodation.

Nader saw Shiʿism as, while in many ways distinct, essentially similar to the four madhhabs of Sunni Islam, and attempted to convince Ottoman diplomats to officially accept it as a fifth. (At the same time, of course, he also tried to convince Shiʿi clerics back home that this was merely a peacekeeping gesture, and that outside of some symbolic concessions nothing would fundamentally change.) This effort was singularly unsuccessful: the Ottomans questioned whether they even had the authority to make such a broad innovation, and regardless were not terribly interested in doing so.

In 1743, Nader Shah tried again, this time organizing a debate between Sunni and Shiʿi clerics in the city of Najaf, in Iraq. Abdullah b. Husayn al-Suwaydi, a Baghdadi cleric who attended the debate as a representative of the Ottoman governor of Iraq, documented the details of this dialogue in a treatise On the Unity of Islamic Theological Schools. The name is telling: unlike the diplomats of 1736, al-Suwaydi was somewhat convinced by the shah's project, even if he did think it would be simpler and preferable to just convert Iran back to Sunnism. Nader hoped to use this momentum to obtain formal recognition of the Jaʿfari (i.e., Shiʿi) maddhab in the Treaty of Kurdan, negotiated with the Ottomans in 1746. While he was again unsuccessful, the Ottomans did grant that Iranian Shiʿites making the hajj could attach themselves to the Syrian caravan: at once a statement of the extent of the separation Ottoman administrators generally tried to enforce between the two sects, and the fact that small concessions were possible in an environment of overall inflexibility. Just as importantly, however, I should stress that these concessions were small: Shiʿism was never, as far as I am aware, recognized as deserving of equivalent autonomy as Judaism or the various Christian sects, much less Sunnism.

Nader Shah's death saw, to some degree, retrenchment on both sides, as his Iranian successors once more adopted Shiʿism as a major element of their rule. Iraq was a particularly problematic area for the Ottomans, as a constant influx of Iranian money and clerics meant the region's loyalties were forever in question. Various methods were proposed and adopted to try and counteract Shiʿi influence, some directly injurious to Shiʿites (the marriage ban for example), others merely intended to uplift Iraqi Sunnis at the expense of their Shiʿi neighbors (many plans for investment in religious education). This tension persisted pretty much until the end of the empire, but it increasingly competed for attention with European colonial encroachment. By the 1890s, Ottoman officials were seriously considering Sunni-Shiʿi unity in a way that would have been unthinkable in the day of Nader Shah. We shouldn't take this as too great a sign of official laxity towards Shiʿis—the proposed pan-Islamic projects were always under the definitively Sunni auspices of the sultan. But in broad strokes, even these modest attempts at rapprochement were great progress from the days when stating that "ʿAli is the friend of God" would have been grounds for deportation to the Balkans, so... progress!