Thank you! Concerning Question 2: thanks for the easy counterexample. Looking at the more complicated $G/B$, we see that $B_i$ is a union of shifted cones indexed by words of length $i$ in the Weyl group. It's this kind of interesting behaviour that I would like to see generalized. I will edit Question 2 to mention this.

Thank you! Regarding Question 2: my $f$ was supposed to depend on $i$. I will edit the question to clarify this.

OOPS! Yes, you're right, thank you! But what if we assume that $X$ and $Y$ have trivial $\text{Pic}^0$, does it help? Is there a good description of the cokernel of $f^*: \text{Pic }X\to \text{Pic }Y$ (or just the next group in some long exact sequence extending $f^*$)?

A. Klyachko, Equivariant vector bundles on toral varieties, Math. USSR Izv. 35 (1990), 337–375.

It is strange then that they neither check if these bundles are toric nor make reference to Klyachko's article.

Concerning the last reference: there is a complete description of toric vector bundles due to Klyachko 1989. In his long paper, among other things, he gives an iff condition for all rank 2 toric vector bundles to be split.

@Mohan: Sorry for being sketchy and giving the wrong answer. Unfortunately, I don't understand your argument. Is my "vector bundle" intuition wrong? Are you saying that $L_a \oplus L_{-a}$ does not depend on $a$?

Say we start with $X=\mathbb{P}^n$. What happens to the Hodge diamond if we blow up a $p$-dimensional linear subspace?

Does the Leray spectral sequence + Lefschetz hyperplane theorem on every fiber help?

I don't think this answer addresses the question very well...