Suppose given a smooth morphism $f:X\to Y$ between varieties over $\mathbb{C}$ whose fibres are $\mathbb{P}^n$. Then I have an equality of Hodge polynomials 
$H(X) = H(Y)H(\mathbb{P}^n)$, say because the hyperplane class generates the cohomology of $\mathbb{P}^n$ and hence $f\_\* \mathbb{Z}\_X$ cannot have monodromy. 



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Is every projective space bundle locally trivial in the Zariski topology?
It is not necessarily trivial in the Zariski topology. Consider for instance the plane quadric $\{x^2+sy^2+tz^2\}\subseteq \mathbb P^2\times\mathrm{Spec}\mathbb C[s,s^{-1},t,t^{-1}]$ as a family of $\mathbb P^1$'s over $\mathrm{Spec}\mathbb C[s,s^{-1},t,t^{-1}]$. It is not even isomorphic to $\mathbb P^1$ over the generic point of 
$\mathrm{Spec}\mathbb C[s,s^{-1},t,t^{-1}]$ as it doesn't have a rational point. As for the multiplicativity in the Grothendieck group there are examples when it is not true not even in the localised and completed Grothendieck group, see arXiv:0903.3143.


