Geodesics: From Affine Connection to Length Minimisation On a manifold $M$ with affine connection $\nabla$, I can define a geodesic as a special integral curve. Namely a geodesic is an integral curve $\gamma$, generated by a vector field $X$, that also parallel transports this tangent vector i.e. it satisfies
$$\tag{$\star$}\nabla_v v =0$$
Note that we are able to define the geodesic without any mention of a metric at all
Q1: does this mean geodesics exist on non-Riemannian manifolds?
Q2: how can I start with the definition $(\star)$, introduce a metric tensor $g$ and show that the geodesic will be the curve that minimises arclength. Everything I read assumes that the connection is the affine connection. I am aware that the metric picks out the affine connection as the only metric compatible connection (satisfying $\nabla g=0$) but that doesn't mean other connections don't exist. Supposing $(\star)$ was defined with respect to a different connection, what happens?

You can use such definition without using a metric, for example see the notion of affine manifold, like the quotient of $\mathbb{R}^n-\{0\}$ by the homothetic map $h(x)=2x$, it is endowed with a connection inherited from the classical flat connection of $\mathbb{R}^n-\{0\}$ since that connection is preserved by $h(x)=2x$.
Geodesic can be defined in Riemannian geometry with the distance. A Riemannian metric on $M$ induces a distance and if $M$ is complete, a geodesic between $x,y$ with will be the path between $x$ and $y$ which is the critical point of function. See the answer here.
Shortest path to a geodesic
