What does a flat yield curve mean? Does it mean it has a modified duration of zero?
It means that the YTMs for bonds of all maturities are all the same.
A yield curve doesn’t have a modified duration, so the second question, as written, doesn’t make sense. If you’re asking whether it means that bonds have a modified duration of zero when the yield curve is flat, the answer is no.
Does this include bonds with embedded options as well?
It depends on which yield curve is being discussed.
Typically, when people use the phrase “the yield curve” without qualification, they mean the (US) Treasury yield curve, which, these days, has essentially only option-free bonds.
You certainly can construct a yield curve for, say, only callable bonds, or only putable bonds, or whatever. But it’s not common.