Jaak Suurpere wrote:

Earth now has just 1 satellite. Moon.

Yet many of the simulations of the Moon-forming impact result in
two moons, of which only one sticks around for long (which hints at
what the long-term outcome might be).

We know that perturbations by Sun do not rule out existence of
satellites for many planets,...

Although I should add there's a very good reason (solar
perturbations) that Mercury & Venus do not have moons. It doesn't
(yet) apply to Earth, but it could for a different solar system

What orbits could a natural satellite of Earth naturally have,
assuming that Moon has orbit as in OTL?

The basic tidal evolution is simple, for a moon in a circular
orbit. If it is below synchronous orbit, tidal effect will collapse
the moons orbit, ending up with the moon spiraling inside the Roche
limit (or even impacting the surface intact - the fate of Phobos).
Outside sychronous orbit, tidal interactions slow the planet's
rotation, expand the moons orbit, until either the system enters tidal
lock (as with Pluto-Charon) or the moon escapes (as would likely
happen with our own Moon, given enough time). Finally, if the moon is
in a retrograde orbit, it will spin down the planet, collapse the
moons orbit, resulting in the moon again entering the Roche limit or
impacting the hapeless planet below (a'la Triton).
As to the effect of the Moon on this system, moons (for reasons I
honestly don't understand fully) can become locked in stable
mean-motion resonances. If, for instance, you had a small moonlet in a
closer orbit (but outside synchronous),it would expand its orbit,
until it entered into a 2:1 resonance (or, perhaps, 3:1? Not sure)
with the Moon, at which point it would (I *suspect*!) become locked
into resonance.

What would the ring do next?

Sit there, for a long period of time. Charging by radiation, and
interactions with the Earth's B-field, might lead to some interesting
"spokes". The inclined Moon orbit might be *very* interesting, perhaps
warping the rings, but here again there could be some funny resonance
issues (not gaps; gaps too are resonant issues, but I'm thinking
beyond that).

[fragments] would then orbit for some time on closeby orbits and
undergo collisions at slow speed. What else can be said about this
process?

Define "slow". The relative speed for two objects formed from tidal
disruption of a single parent would not be "slow enough".

At which geological - or historical - times could the Earth have had
rings or satellites without us knowing about it now?

Geologically-speaking, it could have - but it's very very unlikely
within the last 3 billion years or so.

--
Brian Davis