What was their attitude towards the discovery of the Americas

I've written some about it here

Did they ever have any overseas territories (not necessarily in America)?

I'll try to answer this specific question, sorry if it doesn't satisfy you.

No, However several Muslim states in the Indian Ocean expressed their allegiance to the Ottomans, and the Ottomans themselves also send troops and support for said states to resists European advances or for their affairs, Though Ottoman influences effectively disappeared from these states by the 17th century for a reason or another. While not exactly a colonial venture, these Ottoman ventures are an attempt to unite various Muslim states in the Indian Ocean under the banner of Pan-Islamism led by the Ottoman Sultan as the Padishah of Islam

For example, the Sultanate of Gujarat housed an influential Rumi (Turkish or of Turkish descent) community that maintain ties with the Ottoman state and provide intelligence on the Indian Ocean. It started when Malik Ayaz, a powerful merchant and governor of Diu requested the assistance of the Mamluks to protect Muslim shipping against Portuguese incursions which was accepted in 1507. However, Hussein al-Kurdi, the Mamluk admiral that tasked to fight the Portuguese proved to be an untrustworthy and imperious man who had his own agenda and ambition. By 1509, Malik Ayaz withdrawn his cooperation with the admiral, which ended up destroying the Mamluk fleet after several victories against the Portuguese. A second and larger Mamluk fleet was send in 1515, but rather enganging the Portuguese, Hussein diverted the fleet into Yemen to establish a fiefdom there. Convinced of Mamluk inability to help them, Malik Ayaz then proceeded to ask the Ottomans instead. After the conquest of Egypt, Selim I promised to help the Gujaratis, however an actual fleet would not be sent until the reign of Suleyman I, who send an Ottoman fleet under Mustafa Bayram and arrived at Gujarat on 1531. Immediately after arriving, Ottoman forces brought by the ships (numbered around 600 Ottoman and around 1,000 Arab auxiliaries) routed the Portuguese siege of Diu. After this success, the Sultan of Gujarat rewarded the men and offered them to stay in Gujarat and enter his service. Most of the men accepted, and these emigres will become an influential and important player in Gujarat while maintaining close ties with the Ottoman Empire. However, this relationship will end after the Ottomans failed to recapture Diu from the Portuguese, which were given to the Portuguese by Sultan Bahadur of Gujarat in exchange for immediate military assistance against the Mughals. The Ottomans first besieged Diu on 1538, which was lifted by the Ottoman commander Hadim Suleyman Pasha for an unclear reason even though they nearly succeeded. The second siege started on 1546, and after a bloody battle with both sides nearing the point of exhaustion the Portuguese were saved by a timely arrival of a relief fleet. After this, the formal alliance and Ottoman influence in Gujarat more or less declined or waned, though there are efforts to reinvigorate it and the Rumi community is still eager to retain Ottoman influence in the area. The prestige and respect for the Ottoman dynasty also persisted in the period, as local Muslims recognized Ottoman sultan's position as caliph and hutbes were read in the name of Ottoman sultan

Another example is also Aceh. At first the Acehnese received around 300 Ottoman men on 1538, whether send directly or only a mere windfall from the failed siege of Diu. Nevertheless the Ottoman troops impressed the Sultan of Aceh who later on requested for Ottoman support on 1562. He were granted 10 cannon-founders to aid him casting his own cannons and a vague promise of further help. Later on in 1566, Alauddin Ri'ayat Shah, the Sultan of Aceh send a letter pretty much requesting the Ottoman sultan to annex his lands in exchange of military aid:

We sincerely request that Your Imperial Majesty [the Ottoman Sultan] should no longer consider me, your servant in this land, to be an independent ruler, but instead to accept me as a poor, humble, and downtrodden slave who lives thanks to the charity of your Imperial Majesty, Refuge of the World and Shadow of God [on Earth], in no way different from the governors of Egypt and Yemen or the begs of Jiddah and Aden. . . . With God as my witness, this [city of ] Aceh is one of Your Majesty’s own villages, and I too am one of your servants. Your official Lutfi can personally attest to our circumstances and to our deeds, to the great endeavors we have undertaken for the sake of holy war, and to our firm and sincere longing to enter Your Imperial Majesty’s service.

The Ottoman Sultan Selim II granted the request and sent a fleet of 15 galleys filled with weapons, troops, engineers, and military experts to Sumatra. The fleet was distracted by an insurrection in Yemen, so the Ottomans send another fleet of 2 galleons with similar crews and loadouts which managed to arrive in Aceh, exchanging a shipment of artillery, munitions, gunners, and 500 Ottoman soldiers for a cargo of pepper and other spices. Ultimately, it's this type of commercial transaction rather than alliance of high diplomacy that illustrates Ottoman relationship with Aceh even with the earlier grandiose.

Similar kind of Ottoman influence can be found on other sides of Indian ocean, from Mogadishu to Deccan and even China. The Ottoman Empire during its age of exploration never really attempted to establish a colony, but to unite Muslim states of Indian Ocean under Ottoman banner, which nearly succeeded but eventually passed as the 16th century goes

Sources:

The Ottoman Age of Exploration by Giancarlo Casale

Sixteenth Century Turkish Influence in Western Indonesia by Anthony Reid

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