It's worth pointing out, at the start, that your identity as a "Persian merchant" is in no way a sure sign of your ethnic or religious affiliations! While it's been argued that the reign of Shah ʿAbbās I (1588-1629) saw the replacement of Muslim merchants by non-Muslims (chiefly Armenians) in long-distance trade, Rudi Matthee has pointed out that Armenians had a large stake in the very important silk trade long before ʿAbbās's time, and other minorities like the Persian Jews played commercial roles as well. Even within the defined category of "Muslim Persian merchants," we see internal divisions between various geographic subcultures—Gilaki merchants had their own caravanserais, as did Khorasanis, etc. (It's also worth noting that not all Persian Muslims were Shiʿi, especially closer to the beginning of the sixteenth century.) So while I'll be taking your question as referring to a Shiʿi, Tajik merchant from somewhere near the court (Tabriz or Qazvin), just note that circumstances absolutely could vary.

(On a similar note, you probably wouldn't be settling in Constantinople for business reasons—Erzurum, in Eastern Anatolia, was one entropôt city for most of the trade brought from Iran, as were Smyrna Aleppo and Bursa [thanks for pressuring me here, u/orcasaregreat] further west. But we'll ignore this for the sake of the question.)

Would Persian be easily understood by Ottoman citizens of Constantinople?

At the very least, by anyone who had received a smattering of education! Ottoman artists, intellectuals, and bureaucrats—in general—spoke, or at least wrote, Persian as a pretty standard second language. In fact, the further back in Ottoman texts you go, the more Persian there is! While rural laborers might not be able to understand the language, Persian cultural influence was much more pronounced in urban environments—for this reason the Ottoman Empire is often considered to be part of the "Persianate World," a modern academic term denoting the wide geographical space in which Persian linguistic and cultural influence had a formative effect on local cultures.

In short, Persian acted as something of a koine within this world, which included the Ottoman Empire. One reason for this was the cultural influence of Persian culture more generally; another, perhaps, was the reluctance of Tajiks to learn a language other than Persian except when absolutely necessary. He's speaking of the Timurid domains surrounding Herat, at a rather far distance from the Ottoman Empire, but the Turkic poet and politician Mir ʿAlī Shīr Navāʾī makes a rather clear statement about this near the beginning of the sixteenth century:

There are more literates among the Persians. But although that is true, Turks from notables to commoners and from slaves to lords are acquainted with the Persian language and speak it according to their particular stations. Turkish poets even write beautiful poems in Persian. In contrast, not one member of the Persian nation, be he brigand or notable or scholar, can speak Turkish or understand anyone who does. If one in a hundred or even in a thousand learns and speaks this language, everyone who hears him knows that he is a Persian. With his own tongue he makes himself an object of ridicule.

Of course, merchants in long-distance trade had further pressures to deal with and you might find it easier to use Turkish once in a while; but presuming you're dealing with "the right sort" of people, you would be able to get pretty far with just Persian. (And in international trade, I would expect your contacts to be better acquainted with Persian than the average Constantinopolitan.)

Would Persian culture be compatible with 16th century Ottoman culture?

Largely speaking, yes! This is part and parcel of that "Persianate world" thing I mentioned earlier. Some aspects may have been different—for example, it's still unclear to what extent coffee achieved as rapid an acceptance in Iran as it did in the Ottoman empire (where, by the second half of the century, coffee consumption became a prominent fixture of urban life).^(*) And with the adoption of Hanafism as the official madhhab of the Ottoman empire, depending on your own religious beliefs, you might have seen some differences in what sorts of acts were permissible. But Constantinopolitan society would have felt much more familiar than, say, that of Cairo or Fes.

Depending on when you relocated to Constantinople, you might even find a ready-made circle of Persian émigrés, many of whom moved to the city from Tabriz as a result of the many military campaigns of the sultans Selim I and Süleyman I against the Safavids. According to the Ottoman biographer ʿÂşık Çelebi, this development discomfited Selim somewhat—he his supposed to have complained:

"We arrived in the land of the Persians and forced the talented ones to emigrate to the land of Rum. My goal was that the talented men of Rum would surpass the Persians and achieve high rank in arts and crafts. But I hear that people still consult Persian masters, and that they feel ashamed to consult the Rumis . . . It seems there is no one [among the learned Rumis] who can beat the Persians as I beat the [Safavid] king."

His companion Hayali Çelebi is supposed to have attempted to cheer him up, saying: "Oh my sultan,under the shadow of your felicity, Istanbul is now like Tabriz."

Would I be stigmatised due to my religion/ethnicity?

Ah, now this is probably the most interesting part of the whole question! Short answer: it heavily depends on what part of the sixteenth century we're talking about. While trade between Iran and the Ottoman Empire wasn't entirely stabilized until the Peace of Zuhāb (1639), it was carried on intermittently throughout the century. As we saw above, there were plenty of times when being a Persian was not only fine, but in fact conferred distinct social credit as a representative of a cultured and enlightened people. This is complicated somewhat, however, by the fact that you're not engaged in a cultural profession—and merchants could come under strong suspicion when tensions flared up between the Ottomans and the Safavids.

Things would probably be worst for you if you were in Constantinople around 1511, the year Shah Kulu's rebellion broke out in Anatolia. Shah Kulu and his sympathizers were Qizilbāsh (Tk. Kızılbaş), spiritual as well as political partisans of the Safavids. Sultan Selim I therefore responded with an incredible harshness towards not only declared partisans of the Safavids, but Anatolian Shiʿis in general—an immense campaign of repression involving the arrest and execution of a large number of people whose loyalty to the state was now suspect. As part of this campaign, Selim also arrested a large number of Iranian merchants in Anatolia (remember, as I said before, setting up shop in Istanbul would not have been the normal way of doing business) on the charge of sending military technology and people skilled in its use to Safavid Iran. While this policy was criticized by prominent Ottomans including the future Sultan Süleyman I and the Grand Vizier İbrahim Pasha, the merchants were not released until Süleyman's accession to the throne in 1520. Similar persecution of Iranian merchants took place in the reign of Süleyman's own son Selim II (r. 1566-1574). Is that enough stigma for you?

But the mere fact that these persecutions took place makes clear, I hope, that in ordinary times Persian merchants did not experience too much resistance to engaging in trade within Ottoman borders—at least, not enough to make it not worth the effort.