For question 1, let $L_\alpha$ denote the Levi of the parabolic $P_\alpha$; then $L_\alpha$ has derived group $SL_2$. Let $B_\alpha$ denote the Borel of $L_\alpha$ such that $B \cap L_\alpha = B_\alpha$. Note that $V_{\lambda'} = O_{G/B}(V_\alpha(\lambda'))$, where $V_\alpha(\lambda')$ is the $L_\alpha$ representation induced from $\lambda'$. That is, $V_\alpha(\lambda') = Ind_{B_\alpha}^{L_\alpha} (\chi_{\lambda'}) = Ind_B^{P_\alpha} (\chi_{\lambda'})$, where $\chi_{\lambda'}$ is the 1-dimensional representation corresponding to the weight $\lambda'$. Since $L_\alpha$ has derived group $SL_2$, the weights of $V_\alpha(\lambda')$ are precisely $\lambda'- i \alpha$, $0 \leq i \leq n-1$. Hence $V_\alpha(\lambda')$ has (considered as a $B$-module now by extending trivially over the unipotent radical of $P_\alpha$) a $B$-equivariant filtration with quotients $\chi_{\lambda' - i \alpha}$, $0 \leq i \leq n-1$, so that $V_{\lambda'}$ has the desired $G$-equivariant filtration.

For question 3, the fibers of $\pi'$ are all isomorphic to $L_\alpha / B_\alpha$, the flag variety of $L_\alpha$; and this flag variety is isomorphic to $\mathbb P^1$. Since $(\lambda - \lambda' - \alpha)(\alpha^\vee) = -1$, under the identification of $L_\alpha / B_\alpha$ with $\mathbb P^1$, we see that the restriction of $p'^* O_{G/B}( \lambda - \lambda' - \alpha )$ to the fibers of $\pi'$ has degree -1, as desired. (I assume you want $p'^* O_{G/B}( \lambda - \lambda' - \alpha )$ here and not $p'^* V_{\lambda'}(\lambda - \lambda' - \alpha)$, since $V_{\lambda'}(\lambda - \lambda' - \alpha)$ is not a bundle on $G/B$).

For question 1, let $L_\alpha$ denote the Levi of the parabolic $P_\alpha$; then $L_\alpha$ has derived group $SL_2$. Let $B_\alpha$ denote the Borel of $L_\alpha$ such that $B \cap L_\alpha = B_\alpha$. Note that $V_{\lambda'} = p_\alpha^* O_{G/B}(V_\alpha(\lambda'))$, where $V_\alpha(\lambda')$ is the $L_\alpha$ representation induced from $\lambda'$. That is, $V_\alpha(\lambda') = Ind_{B_\alpha}^{L_\alpha} (\chi_{\lambda'}) = Ind_B^{P_\alpha} (\chi_{\lambda'})$, where $\chi_{\lambda'}$ is the 1-dimensional representation corresponding to the weight $\lambda'$. Since $L_\alpha$ has derived group $SL_2$, the weights of $V_\alpha(\lambda')$ are precisely $\lambda'- i \alpha$, $0 \leq i \leq n-1$. Hence $V_\alpha(\lambda')$ has (considered as a $B$-module now by extending trivially over the unipotent radical of $P_\alpha$) a $B$-equivariant filtration with quotients $\chi_{\lambda' - i \alpha}$, $0 \leq i \leq n-1$, so that $V_{\lambda'}$ has the desired $G$-equivariant filtration.

For question 3, the fibers of $\pi'$ are all isomorphic to $L_\alpha / B_\alpha$, the flag variety of $L_\alpha$; and this flag variety is isomorphic to $\mathbb P^1$. Since $(\lambda - \lambda' - \alpha)(\alpha^\vee) = -1$, under the identification of $L_\alpha / B_\alpha$ with $\mathbb P^1$, we see that the restriction of $p'^* O_{G/B}( \lambda - \lambda' - \alpha )$ to the fibers of $\pi'$ has degree -1, as desired. (I assume you want $p'^* O_{G/B}( \lambda - \lambda' - \alpha )$ here and not $p'^* V_{\lambda'}(\lambda - \lambda' - \alpha)$, since $V_{\lambda'}(\lambda - \lambda' - \alpha)$ is not a bundle on $G/B$).

For question 1, let $L_\alpha$ denote the Levi of the parabolic $P_\alpha$; then $L_\alpha$ has derived group $SL_2$. Let $B_\alpha$ denote the Borel of $L_\alpha$ such that $B \cap L_\alpha = B_\alpha$. Note that $V_{\lambda'} = O_{G/B}(V_\alpha(\lambda'))$, where $V_\alpha(\lambda')$ is the $L_\alpha$ representation induced from $\lambda'$. That is, $V_\alpha(\lambda') = Ind_{B_\alpha}^{L_\alpha} (\chi_{\lambda'}) = Ind_B^{P_\alpha} (\chi_{\lambda'})$, where $\chi_{\lambda'}$ is the 1-dimensional representation corresponding to the weight $\lambda'$. Since $L_\alpha$ has derived group $SL_2$, the weights of $V_\alpha(\lambda')$ are precisely $\lambda'- i \alpha$, $0 \leq i \leq n-1$. Hence $V_\alpha(\lambda')$ has (considered as a $B$-module now by extending trivially over the unipotent radical of $P_\alpha$) a $B$-equivariant filtration with quotients $\chi_{\lambda' - i \alpha}$, $0 \leq i \leq n-1$, so that $V_{\lambda'}$ has the desired $G$-equivariant filtration.

For question 1, let $L_\alpha$ denote the Levi of the parabolic $P_\alpha$; then $L_\alpha$ has derived group $SL_2$. Let $B_\alpha$ denote the Borel of $L_\alpha$ such that $B \cap L_\alpha = B_\alpha$. Note that $V_{\lambda'} = O_{G/B}(V_\alpha(\lambda'))$, where $V_\alpha(\lambda')$ is the $L_\alpha$ representation induced from $\lambda'$. That is, $V_\alpha(\lambda') = Ind_{B_\alpha}^{L_\alpha} (\chi_{\lambda'}) = Ind_B^{P_\alpha} (\chi_{\lambda'})$, where $\chi_{\lambda'}$ is the 1-dimensional representation corresponding to the weight $\lambda'$. Since $L_\alpha$ has derived group $SL_2$, the weights of $V_\alpha(\lambda')$ are precisely $\lambda'- i \alpha$, $0 \leq i \leq n-1$. Hence $V_\alpha(\lambda')$ has (considered as a $B$-module now by extending trivially over the unipotent radical of $P_\alpha$) a $B$-equivariant filtration with quotients $\chi_{\lambda' - i \alpha}$, $0 \leq i \leq n-1$, so that $V_{\lambda'}$ has the desired $G$-equivariant filtration.

For question 3, the fibers of $\pi'$ are all isomorphic to $L_\alpha / B_\alpha$, the flag variety of $L_\alpha$; and this flag variety is isomorphic to $\mathbb P^1$. Since $(\lambda - \lambda' - \alpha)(\alpha^\vee) = -1$, under the identification of $L_\alpha / B_\alpha$ with $\mathbb P^1$, we see that the restriction of $p'^* O_{G/B}( \lambda - \lambda' - \alpha )$ to the fibers of $\pi'$ has degree -1, as desired. (I assume you want $p'^* O_{G/B}( \lambda - \lambda' - \alpha )$ here and not $p'^* V_{\lambda'}(\lambda - \lambda' - \alpha)$, since $V_{\lambda'}(\lambda - \lambda' - \alpha)$ is not a bundle on $G/B$).