In the equatorial system, the star is located by right ascension (alpha) and declination (delta), Alpha is measured in hours east from the vernal equinox, delta in degrees north or south of the equator. This is convenient for telescope users in finding objects and pointing their instruments.

In the ecliptic system, coordinates are given in terms of ecliptic longitude (lambda), measured in degrees east from the vernal equinox and latitude (beta) measured in degrees north or south of the ecliptic. This coordinate system is useful for calculations of planetary motion and in spacecraft navigation.

The galactic coordinate system is also measured in degrees of longitude east of the galactic center (l, in the direction of Sagittarius) and degrees of latitude north or south of the galactic plane (b, as defined by the Milky Way). Galactic coordinates are useful in galactic studies.

But beware, all three of these coordinate systems have their origin, or zero point, at the earth. All three of these coordinate systems define locations on an imaginary sphere with the earth at its center. In other words, the ecliptic coordinate system is not centered at the sun, and the galactic system does not have its origin in the galactic core. The sun is in the galactic disk, about 30000 light years from the galactic core, or about 1/3 of a galactic diameter.

Several bodies in the solar system probably actually did this, some of the retrograde moons in the outer planets come to mind, but this is probably a result of a collision long ago that knocked a planet or satellite over on its side or even shattered it into fragments and sent stuff flying around chaotically. In the resulting confusion, several of the pieces interacted in just the right way and was “captured”. In general, interacting bodies either collide, or are ejected away from each other, the conditions have to be just right for them to wind up in orbit, and it usually requires an external boost or nudge to balance the energy and momentum books.

As for the retrograde rotation of Venus and Mercury, this is due to tidal forces caused by gravitational interaction with the Sun, similar to the way the moon is tidally locked onto the earth, so the same side is always facing us. Its a different mechanism all together

I’m not saying that capture is impossible, just highly unlikely. In the early solar nebula, there was all sorts of confusion, tangled orbits, collisions, ejections, and stuff bouncing around. No doubt some captures took place, but they could easily be identified today because of their retrograde, highly elliptical orbits and backwards rotation. But in general, the vast number of solar system objects revolve AND rotate in a counter-clockwise direction (as viewed from the north ecliptic pole). This is a consequence of the original rotation of the protoplanetary disk. After the system stabilized, subsequent collisions may have knocked over or spun backwards some rotating bodies, but they are few and far between, and most are small planetoids and Oort Cloud and Kuiper Belt objects.

Now consider the sun flying around the orbital path of its great revolution around the galactic nucleus, it, and its surrounding disc stars are flying together like a cloud of buckshot. If this stream of stars suddenly plows into another star stream left behind in the wake of some colliding dwarf galaxy in the past, the sun and any of the stars or debris in that stream will be approaching at colossal relative velocities (even though their velocity relative to the galactic center may not be all that high. A capture is highly unlikely.

Astronomers have spent a lot of time studying the “proper motion” of stars, that is, their motion relative to the sun.
We can measure the line-of-sight velocities with spectrometers (looking for red and blue shifts) and their velocities across the line of sight by carefully measuring the motion of their images on photographic plates taken decades apart. The studies have shown two distinct populations of stars, the disc stars rotating along the disc with the sun, and other stars that belong to a population of randomly distributed orbits around the galactic center. This tells us that when the galaxy first started off as a roughly spherical cloud of gas, the “halo” stars formed in a huge spherical volume. After the gas disc flattened out due to its rotation, stars formed later remained in the plane of the galaxy. From this point of view, we see the sun’s disc population companions traveling along with us at relatively low velocities, say ,several tens of km/sec (relative to us). The halo stars flying through the disc right now from all over the place, above and below the disc seem to be moving at hundreds of km/sec. Actually all these stars are moving at approximately the same speeds (relative to the galactic center) but the ones floating along with us in the stream appear much slower because we’re moving along with them.

Again, that Wikipedia article on Stellar Populations covers this material very well. The fact that stars could be divided into distinct populations due to their kinematic behavior, and that this was highly correlated with their chemical composition as determined through spectroscopy, gives us clues as to just how the galaxy formed billions of years ago.

I now have to wonder how many of the sun’s more distant orbiters are actual Oort bits, or stuff picked up in one of these transits:

2015 GT50
uo5m93
2013 FT28
P9
2015 BP519
2010 GB174
Sedna
2015 RX245
2012 VP113
2004 VN112
2014 SR349
2013 RF98
2007 TG422
2013 SY99
2015 TG387
2015 KG163
2014 FE72

…with more being found every year…

The second supports his six-gun and holster, and it is at a 23.5 degree angle to the equator. We call that the “obliquity of the ecliptic”, or the tilt of the earth’s rotational axis to its orbital path of revolution about the sun. The places where the ecliptic and equator cross, where the belts coincide at the equinoxes, are where the sun crosses the equator heading north in March and south in September.These occur traditionally (according to the astrologers) in the constellations of Aries and Libra, although precession has shifted them along to Pisces and Virgo.

The third belt keeps the pistolero’s fanny pack securely lashed to his ass. It crosses his pants belt at an angle of about 60 degrees, and corresponds to the path in the sky of the Galactic Equator, or what we call the milky way.

Ecliptic and equator are related, the Sun and most of the planets and satellites rotate and revolve in roughly the same plane, with only a few exceptions, as you point out. This is no doubt an artifact of the rotation of the original nebula that formed the solar system. But the plane of revolution of the solar system, as well as the plane of rotation each of the stars, have nothing to do with the rotational motion of the galaxy. The rotations about their axes of the individual stars are totally uncorrelated and unrelated to the rotation of the galaxy. In fact, they appear to be completely random. Studies of binary stars and of stellar rotations indicate that the axes of rotation are distributed randomly, with no preference to the rotation of the galaxy.

As the solar system flies around the galaxy, the plane of the planets’ revolution (the ecliptic) is tilted relative to the direction of travel by about 40 degrees.