In Turkey or any country there are no police at the border requiring you to identify your gender preference and attaching a big orange label to your clothes if you identify one way or the other. You could enjoy any city in the world as a single male traveler, there is no need to intentionally label yourself beyond that unless visiting nightclubs and bars at night in search of companionship. If you were traveling as a couple then I could understand a bit more concern but being single, no worries.
As for your other questions, Istanbul is pretty easy to navigate, they use the same alphabet so even signs not in English can be figured out (unlike where you're headed in Asia) but the metro and tram are clearly labeled and ticket machines are multi lingual. Your best bet if using the metro from the airport is to buy the Istanbul Karte from one of the vending machines at the airport metro station, they are much harder to find in the city. It'll allow you discounted fares on Metro, Tram, and Bus when using the prepaid card but also many buses are cashless so have to have a card to travel on those buses. There is usually one dedicated machine that sells them, the metro ticket machines also say they sell them but have never gotten one to work. As for spoken language, in the tourist areas, you'll find that almost all Turks speak English and perhaps other popular languages to get people into their business. Outside the tourist areas though, I've still managed to get by with basic communication with people in Turkey with a combination of gestures, body language, and English. Language barriers don't have to be as high as people make them to be, a smile and making an effort goes a very long way.
You'll have enough time to see some sites in the old city, that's where the majority of the key sites are concentrated, though the ones with ticket offices won't be open quite as early. If you go to the Hagia Sophia get your ticket early as there can be quite a queue once the tour groups start arriving. There are plenty of restaurants in the tourist districts of the old city to grab breakfast while you're waiting for stuff to open up.
Finally enjoy Istanbul, it's a beautiful city with great food around every corner.