  Let $M$ be a metric space and $X$ be a Banach space. In this paper we address
several questions about the structure of $\mathcal F(M)\widehat{\otimes}_\pi X$
and $\mathop{Lip}(M,X)$. Our results are the following:
  (1) We prove that if $M$ is a length metric space then $\mathop{Lip}(M,X)$
has the Daugavet property. As a consequence, if $M$ is length we obtain that
$\mathcal F(M)\widehat{\otimes}_\pi X$ has the Daugavet property. This gives an
affirmative answer to [13,Question 1] (also asked in [24,Remark 3.8]).
  (2) We prove that if $M$ is a non-uniformly discrete metric space or an
unbounded metric space then the norm of $\mathcal F(M)\widehat{\otimes}_\pi X$
is octahedral, which solves [6, Question 3.2 (1)].
  (3) We characterise all the Banach spaces $X$ such that $L(X,Y)$ is
octahedral for every Banach space $Y$, which solves a question by Johann
Langemets.
