Is it safe for touring in Beirut being an American this time of year? Would I be stopped at the airport?
Do I need a Lebanese passport to travel to Beirut along with my American passport?
Unless you are of Lebanese descent you cannot obtain a Lebanese passport. If you are eligible for a Lebanese passport, be careful if you use it to enter the country: Lebanese citizens are subject to compulsory conscription - you might find yourself in the army!
There are certain parts of Beirut in which trouble occurs from time to time - locals will tell you where they are and when it is a good time to stay away.
The US State Dept warning on Lebanon is pretty stark: http://travel.state.gov/content/passports/en/alertswarnings/lebanon-travel-warning.html
There's no reason to expect your US citizenship will single you out for attention at the airport.
https://www.worldnomads.com/travel-safety/lebanon/lebanon-obeying-the-local-laws-and-getting-around-safely about 2 years ago