Even today, calling the situation in Benghazi "relatively safe" can only be a stretch. Despite frequent statements by various Libyan politicians from the East, as well as by representatives of the House of Representatives in Tobruk, that the situation in the city is under their control, this is far from always the case. No "government" of the country is ready to guarantee the safety of foreigners. However, Qatar Airways, Libyan Airlines, Gulf Air, Turkish Airlines currently have flights to Benghazi.
No, I found it. Hopefully the link will work fine. In general, the English Wikipedia says that it works: As of December 2015, Air Libya operated out of Benina International Airport and operates flights to Tripoli, Alexandria, Kufra and Tunis while Libyan Airlines operates to Amman and Kufra.