I don't have a complete answer, but let me just note that for a regular dominant weight $\lambda$ your question 2) would be true if there is a Frobenius splitting of the cotangent bundle $T^*_{X^n}$ of $X^n$ which compatibly splits the diagonal copy $\Delta$ of $T^*_X$, the cotangent bundle of $X$. Since $T^*_{X^n}$ is the cotangent bundle of the flag variety of the semisimple algebraic group $G^n$, there is a candidate for this splitting, namely the splitting of Kumar, Lauritzen, and Thomsen that you mention. I don't know, though, if their splitting compatibly splits $\Delta$; that is an interesting question.

I don't have a complete answer, but let me just note that your question 2) would be true if there is a Frobenius splitting of the cotangent bundle $T^*_{X^n}$ of $X^n$ which compatibly splits the diagonal copy $\Delta$ of $T^*_X$, the cotangent bundle of $X$. Since $T^*_{X^n}$ is the cotangent bundle of the flag variety of the semisimple algebraic group $G^n$, there is a candidate for this splitting, namely the splitting of Kumar, Lauritzen, and Thomsen that you mention. I don't know, though, if their splitting compatibly splits $\Delta$; that is an interesting question.