Concerning the first question.
No there are geodesic spaces that admit smooth families of dilatations but are not made of flat Banach metrics. In particular, some Carnot-Caratheodory spaces are.
For example, consider the Heisenberg Lie group $H$, which can be thought of as $\mathbb R^3$ equipped with the following group law: $$ (x,y,z)\cdot(x',y',z') = (x+x',y+y',z+z'+x'y) . $$ Observe that for every $t\in\mathbb R$, the map $\phi_t:(x,y,z)\mapsto (e^tx,e^ty,e^{2t}z)$ is a group homomorphism, and these maps form a smooth 1-parameter grooup of diffeomorphisms (and hence a flow generated by a smooth vector field).
Consider a left-invariant two-dimensional distribution $V\subset TH$ spanned by left-invariant vector fields $X$ and $Y$ whose values at $(0,0,0)$ equal $\partial/\partial x$ and $\partial/\partial y$. Equip this distribution with a left-invariant Euclidean metric. The distribution is completely non-integrable, so we get a Carnot-Caratheodory metric on $H$. Observe that $\phi_t$ maps $X$ to $e^tX$ and $Y$ to $e^tY$, hence it is a $e^t$-dilatation of the Carnot-Caratheodory metric.
But the Carnot-Caratheodory metric is very different from Banach metrics. For example, its Hausdorff dimension equals 4.