I finally stumbled upon the term radar to describe this use of a reference timelike worldline. A good starting resource is: Perlick, Volker. "On the radar method in general-relativistic spacetimes.", which points to a bunch of other resources.
The $r^\star$ and $\tau^\star$ are in fact the radar distance and radar time resp.
I still haven't found any name for the surface, though radar bubble suggests itself. There is a region around the reference worldline where these bubbles are globally spacelike and topologically $S^2$ (a bubble). In flat space this region is the entire space, though a gravitating object will make bubbles self-intersect at certain distance.
A radar bubble can fail to be even locally spacelike: if a null geodesic intersects the worldline twice, then the corresponding bubble would include that geodesic. A black hole can do this. It can act as a gravitational mirror, and bounce light coming in on a certain angle from a distant source back.
Addendum
Instead of the light cones themselves, the above also works using the boundary of the chronological future (past). The chronological future (past) of an event $p$ are the events that can be reached from $p$ (can reach $p$) by a timelike path. They are designated $I^\pm(p)$, and their boundaries $\delta I^\pm(p)$.
In a globally hyperbolic spacetime, these are subsets of the past or future light cone, excluding the parts where it self-intersects.
The intersection of a future and past boundary (not to be confused with the boundary of the intersection) $\delta I^+(p) \cap \delta I^-(q)$, is indeed globally spacelike, though not always topologically $S^2$.
Given a timelike path $p(\lambda)$, let's label the bubble $B_p(\lambda,\mu) = \delta I^+(p(\lambda-\mu)) \cap \delta I^-(p(\lambda+\mu))$. Then for a given $\lambda$, the union of bubbles $\bigcup_{\mu \ge 0} B_p(\lambda,\mu)$ is a globally spacelike 3-D hypersurface. This family of hypersurfaces indexed by $\lambda$ foliates the part of the spacetime reachable from the path (unlike the "instantaneous simultaneous spaces" -- even in special relativity, two instantaneous simultaneous spaces at different events of an accelerated worldline will intersect each other).
In particular this is true if the path parameter is just the proper time along the path (i.e. $\tau^\star = \lambda$ and $r^\star = \mu c$).
First we sketch a proof that a timelike geodesic is a maximum of proper time. (We exclude saddle points for now.) Let $\gamma$ be a curve satisfying the geodesic equation, i.e. it is an extremum of proper time defined by $\tau[\gamma]:=\int\sqrt{-\langle\dot\gamma,\dot\gamma\rangle}\,\mathrm{d}t$. It is fairly simple to show that there always exists a curve $\mu$ for which $\tau[\mu]<\tau[\gamma]$, implying $\gamma$ is not a minimum. Construct along $\gamma$ a "tube" which is arbitrarily wide. Let $\mu$ be a curve which has the same start and end points as $\gamma$. Let $\mu$ be confined to the tube along $\gamma$. Now wind $\mu$ along the tube so that it is almost null, i.e. the curve's tangent approaches the null cone at every point on the tube. Thus we have constructed a curve with $\tau[\mu]$ arbitrarily close to zero, which can be made less than $\tau[\gamma]$.
This implies that a geodesic is not a minimum, but cannot determine that a timelike geodesic is not a saddle. However, this is not entirely true either. Here we quote Theorem 9.9.3 in [1]$^1$.
Let $\gamma$ be a smooth timelike curve connecting two points $p,q$. Then the necessary and sufficient condition that $\gamma$ locally maximize the proper time between $p$ and $q$ over smooth one parameter variations is that $\gamma$ be a geodesic with no point conjugate to $p$ between $p$ and $q$.
So a timelike geodesic is not necessarily a maximum of proper time. The study of geodesics does tie in to causal structure, Refs. [1] and [2] are highly recommended for this purpose.
Two standard references on causal structure are:
[1] R.M. Wald, General Relativity (1984).
[2] S.W. Hawking & G.F.R. Ellis, The large scale structure of space-time (1973).
$^1$This is in turn quoted from Proposition 4.5.8 in [2], but I prefer [1]'s wording. Note that the full proof is found in [2].
Best Answer
Spacelike, null and timelike geodesics correspond to geodesics (in a spacetime with signature $-+++$) with a tangent vector $u$ of positive, zero or negative norm, $|u| = g_{\mu\nu} u^\mu u^\nu$. It does correspond to non-accelerated motion, in fact acceleration in general relativity is deviation from geodesic behaviour :
$$a^\mu = \ddot x^\mu + {\Gamma^\mu}_{\alpha \beta} \dot x^\alpha \dot x^\beta$$
Particles moving on those curves are particles of $p^2 = -M^2 < 0$ for timelike curves (generally called massive particles or tardyons), $p^2 = - M^2 = 0$ for null curves (generally called massless particles or luxons), or $p^2 = -M^2 > 0$ for spacelike curves (generally called tachyons).
Massive particles do not necessarily end at timelike infinity, but realistically they do. The only class of particles that can reach null infinity are constantly accelerated ones, such as Rindler observers.