The example of Alex above is a special case of the Borel-Weil-Bott Theorem which gives the following:

Let $G$ be a semisimple complex Lie group with Weyl Group $W$. If for no $w \in W$ we have $w\ast \lambda$ dominant then $H^i(G/B, L_\lambda) = 0 $ for all $i$.

The $L_\lambda$ are all line bundles over $G/B$ which are not necessarily zero, giving a negative answer to one of your questions above.

The example of Alex above is a special case of the Borel-Weil-Bott Theorem applied to $\operatorname{SL}_2/B = \Bbb{P}^1$ with $B$ the standard Borel subgroup in $\operatorname{SL}_2$. The general case is this:

Let $G$ be a semisimple complex Lie group with Weyl Group $W$. If for no $w \in W$ we have $w\ast \lambda$ dominant then $H^i(G/B, L_\lambda) = 0 $ for all $i$.

The $L_\lambda$ are all line bundles over $G/B$ which are not necessarily zero, giving a negative answer to one of your questions above.