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Is it possible to have satellites (natural or not) orbit the same celestial object in different directions, or is the orbital direction dependent on the celestial object's spin?
Also, is the direction of spin of a celestial object dependent on how it orbits a bigger celestial object?
It is absolutely possible. Moons that formed with their planet will be in prograde orbits, but moons that are captured bodies (such as the outer satellites of Jupiter, Saturn, Uranus, and Neptune (Triton's a huge example here)) can wind up in basically any orbital inclination initially.
The Apollo orbits over the Moon were retrograde. A large number of satellites in low Earth orbit are retrograde. The prototypical periodic comet, 1P/Halley, is in a retrograde orbit.
Over the very long term, under significant tidal interactions, retrograde orbits tend to be cranked inward (And Triton will tear away or tear apart all of Neptune's interior moons before Neptune shreds it in the far distant future), but for more distant object, or shorter timescales, they're fine.
Partial and supplemental answer.
As an example1 from Wikipedia's List of natural satellites; Mooons by primary:
(Jupiter's) 84 known irregular moons are organized into two categories: prograde and retrograde. The prograde satellites consist of the Himalia group and three others in groups of one. The retrograde moons are grouped into the Carme, Ananke and Pasiphae groups.
I would like to be able to address the directions that the retrograde moons turn around their own axis of rotation to see if they are all also retrograde rotators2 or not, but I don't even know where to begin.
It could be that for most of these it's simply not known.
1Conveniently, Wikipedia has a Category page Moons with a retrograde orbit which lists both pages of named groups of retrograde moons (including the three in the block quote above) and a list of 136 individual pages for retrograde moons in the solar system!
2Note that if their (retrograde) rotation period is equal to their (retrograde) orbital period, that would make them tidally locked in the same way that our own tidally locked Moon's (prograde) rotation and (prograde) orbital periods are equal.
One full rotation per complete orbit = "locked".
Also note that several tables in Wikipedia give the orbital periods for retrograde moons as negative numbers. This is cute but the more traditional, scientific and accurate if also cumbersome way to do this would be to list the periods as positive number with either a footnote for "retrograde" or to include a column with orbital inclinations which will be near ±180°.
Negative periods suggest time is going backwards, which reminds us of Star Trek's original series episode "Tomorrow is Yesterday" when the Enterprise's chronometers started going backwards. Now Mr. Sulu!
Wikipedia has some examples of satellites in a retrograde Earth-Centric orbit:
Israel has successfully launched seven Ofeq satellites in retrograde orbit aboard a Shavit launcher. These reconnaissance satellites complete one Earth orbit every 90 minutes and initially make about six daylight passes per day over Israel and the surrounding countries, though this optimal Sun-synchronized orbit degrades after several months. They were launched in retrograde orbit so that launch debris would land in the Mediterranean Sea, and not on populated neighboring countries on an eastward flight path.[6][7]
The United States launched two Future Imagery Architecture (FIA) radar satellites into 122° inclined retrograde orbits in 2010 and 2012. The use of a retrograde orbit suggest that these satellites use synthetic aperture radar.[3]
Earth-observing satellites may also be launched into a sun-synchronous orbit, which is slightly retrograde.[8] This is typically done in order to keep a constant surface illumination angle, which is useful for observations in the visible or infrared spectrum's. SEASAT and ERS-1 are examples of satellites launched into sun-synchronous orbits for this reason.
Typically, we launch satellites in a prograde orbit due to the fact that the delta-v needed to reach orbital velocity is lower as the angular velocity of the spacecraft doesn't start at zero, it starts at the same as that of the Earth. If you want to launch into a retrograde orbit, you need to overcome the rotation of the Earth and then have the extra delta-v to get you into orbit.
As for natural satellites, they typically orbit their celestial body in the same direction as that body's rotation due to the fact that they're generally formed from accretion discs around that body. Those accretion discs typically rotate in the same direction as the body because that body itself was typically formed from the same accretion disc/nebula (when it was younger and larger) and angular momentum must be conserved. For something natural to be orbiting a celestial body in a retrograde fashion, it would have to have formed in a non-typical manner (such as from a catastrophic collision) or have been captured from interstellar space. In the former, its new orbit would be a direct result of the forces involved in the collision, in the latter it would be determined by the approach vector of the captured body.
