The planet Theia formed near or at the L4 or L5 Lagrange point of the Earth.
I see that the OP rules out planets that are very close to each other.
I am also not interested in co-orbiting planets, such as anything at
another planet's L4 or L5 point, or dual-planet pairs (not entirely
unlike our Pluto and Charon).
I am not sure why these are excluded. Planetary formation is planetary formation, so I am going to answer with the Theia example anyway. If two planets can form that close to each other it would seem to me to set the inner limit of how close to one another planets can possibly form. The linked wikipedia article states Charon is a big moon that was knocked off of Pluto by an ancient impact much as our own moon; not really relevant to the question of planet formation.
Theia was an ancient planet the size of Mars which collided with and merged with Earth, in the process forming the moon. I here assert that Theia formed from the protoplanetary disk at the L4 or L5 point of the ancient Earth.
Prior to merging with the Earth, Theia is thought to have orbited at L4 or L5.
Theia is thought to have orbited in the L4 or L5 configuration
presented by the Earth–Sun system, where it would tend to remain. In
that case, it would have grown, potentially to a size comparable to
Mars. Gravitational perturbations by Venus could have eventually put
it onto a collision course with the Earth.
Theia hit Earth gently. It was not swooping in from distant reaches of the solar system.
In astronomical terms, the impact would have been of moderate
velocity. Theia is thought to have struck the Earth at an oblique
angle when the Earth was nearly fully formed. Computer simulations of
this "late-impact" scenario suggest an impact angle of about 45° and
an initial impactor velocity below 4 km/s.
That makes sense if it was only at L4 before impact and eventually had its orbit perturbed by Jupiter or Venus.
Earths Titanium Twin
By contrast, Zhang et al.1 find that the Earth and the Moon are
identical in their titanium isotopic compositions within errors of
0.0004% — almost the limit of detectability. This is not the first time the giant impact hypothesis has been challenged by isotopes.
During the past decade, similarities between lunar and terrestrial
rocks have been identified for oxygen6, silicon7, chromium8 and
tungsten9 isotopes. The latter three can be brought into accordance
with the latest giant impact simulations5, if one assumes that Theia
had a composition similar to Mars — possibly the only surviving
planetary embryo from which the larger terrestrial planets accreted10.
However, the oxygen isotopic compositions of terrestrial and lunar
rocks are so similar that, if Theia had a Mars-like composition, it
cannot have contributed more than a few per cent of material to the
Moon-forming disk6. Zhang et al. demonstrate that titanium isotopes
are similarly constraining.
The Earth and the Moon have the same isotopic composition. Where is the contribution from Theia? It is there. The Earth, the Moon and Theia all had the same composition.
On the origin and composition of Theia
If the FeO content of the Earth and the Moon is indeed inherited from
the proto-Earth and Theia, then by implication Theia must have had an
Earth-like isotopic composition (similar to enstatite chondrites,
aubrites or other Earth-like materials like NWA 5400). This is
possible if both the Earth and Theia, but not Mars, were part of an
early inner disk uniform reservoir (IDUR; Dauphas et al., 2014), or if
the inner disk region has been isotopically homogenized in the time
between the isolation of Mars from the disk and the Giant Impact that
formed the Moon.
How close to each other can Earth-mass planets plausibly form from the
Theia formed right next to Earth from materials in the same region of the disk. The near-identical isotopic composition of the Earth and Moon precludes the possibility of Theia being some weird asteroid, or even Marslike in composition. Theia is formed of Earthlike materials. It formed closer to Earth than Mars is. I cannot see how two planets can form any closer than that.
Nothing is going to form closer than the Lagrange points because it would fall into the planet. The example of Theia and Earth forming next to each other sets the minimum distance from one another that planets can form from the protoplanetary disk.