So, first I booked an 11-day "Fire of Anatolia" tour from Fez Travel, a Turkish company. Then I found a 6-day Lebanon tour from Original World Tours, which started the day after my Turkey tour ended. I then arranged flights from RDU-Istanbul (via London) on American and British Airways, Istanbul-Beirut on Middle East Airlines (MEA), and Beirut-RDU (via London and Philadelphia), also on British Airways and American.
Istanbul and Beirut are highlighted in green on the map below.
For the Turkey tour, I was in a group with 8 other people, who were from Sri Lanka, India, Kenya, Poland, and the United States. The tour started in Istanbul with stops in Ankara, Cappadocia, Konya, Pammuke, Ephesus, Gallipoli, and ending in Istanbul. The tour took place a couple of days after the US took its troops out of Syria, causing Turkish troops to go to the Syrian border. However, we did not go near the border with Syria.
For the Lebanon tour, there were only 3 of us; the other two (both Americans) taking the tour as a pre-tour extension to a tour in Saudi Arabia.
SPOILER ALERT: Despite the fact that I was in Turkey just after U.S troops left Syria (and the Turks moved in), and was in Lebanon during the anti-government demonstrations, I felt perfectly safe. In Lebanon, we had to make a number of diversions for our planned itinerary because of demonstrators blocking the roads. But if you're comfortable with lots of military people with rifles walking around (as I have become in my travels), there was really nothing to worry about.
Next: Days 1-3: Istanbul