Geodesically complete implies exponential map defined on all of $T_pM$ This should be completely straightforward as every book I have read on the Hopf-Rinow theorem states this is "obvious". But I can find the life of me to justify this. 
Geodesically complete means every geodesic $\gamma$ extends to all of time $\mathbb{R}$. How is that related to the velocity $\gamma'$ (which hence relates to exp)? If the curve extends all time, then are they implying the speed also extends all time? Does that even make sense since speed is constant for a geodesic, so why would that be attached to the time parameter?

Let $(M,g)$ a Riemannian manifold. Recall that for all $p\in M$ and $v \in T_pM$ there exists some $\varepsilon > 0$ and a $C^\infty$curve $\gamma(t)\colon (-\varepsilon, \varepsilon)\to M$ such that $\gamma$ is the unique geodesic passing through $p$ at time zero with velocity $\dot{\gamma}(0) = v$. The fact that the exponential map is defined for some $v \in T_pM$ has nothing to do with existence of a geodesic through $p$ with velocity $v$ (this always happens!), but with the fact that $\gamma(1)$ is defined ($\varepsilon > 1$). One has to be careful in thinking that a reparameterization would change the speed of a geodesic. A much careful treatment is given in the first five pages of Chapter 3 in do Carmo's book 
