Is it true that there are no projective curves which are also flag manifolds? If so, why?
The projective line is both a curve of genus $0$ and a flag variety for $SL_2$. This is the only example. This is true for about a zillion reasons:
Flag varieties are rational (because of the Bruhat decomposition.) Curves, of genus $>0$, are not.
Flag varieties have transitive group actions. Curves of genus $\geq 2$ do not (see my answer here.)
If you look at the classification of flag varieties, there are only finitely many of any given dimension. In paritcular, there is only one example of dimension $1$.
According to this:
"Flag varieties are naturally projective varieties."