There's a few ways to do it (and the press release is surprisingly good and mentions most of them), but certainly one of the more obvious explanations is that the emitting object, is in a binary system with some other object, and that modulates the emission mechanism somehow.
I don't think either needs to be a black hole though. What I consider the most plausible candidate for emission is probably magnetars (i.e. neutron stars with extremely strong magnetic fields), and there's no reason the companion would need to be a black hole either, you just need a companion that gives a 16 day orbit and that's very easy to do with another star or neutron star.
Variable stars, something involving a cyclic phase transition (from silent to active and back). Something like the Type Ia supernova, where the dense white dwarf siphons off mass from its companion until it reaches the Chandrasekhar limit and bang.
Maybe, but perhaps not in the way you think. A source that emits radiation only at certain angles (think a stellar disco ball) would only "flash" us infrequently as we moved relative to it.
I don't think either needs to be a black hole though. What I consider the most plausible candidate for emission is probably magnetars (i.e. neutron stars with extremely strong magnetic fields), and there's no reason the companion would need to be a black hole either, you just need a companion that gives a 16 day orbit and that's very easy to do with another star or neutron star.