You're right, it's basically because of angular momentum. In essence, if you start with a self-gravitating cloud of material or collection of particles with a mean angular momentum (which needn't be particularly large), then the material smears itself out perpendicular along the plane of rotation (perpendicular to the axis of rotation). The individual motions perpendicular to the plane roughly cancel out through assorted interactions. But, at least on average, the material is all orbiting in roughly the same direction, so that component is preserved. In even broader terms, the evolution allows energy to be lost (through collisions, heating, etc.) but losing angular momentum is much more difficult.
If it were the other way round (i.e. losing energy is difficult, angular momentum easy) then we might expect spherical clouds. For example, in dark matter halos, it's very difficult to lose energy because dark matter cannot radiate energy away, so they remain diffuse and more broadly distributed. i.e. they don't collapse into discs. Giant ellipticals are thought to be the remnants of mergers between massive galaxies and the random orientations of the input angular momenta mean that the remnant has a smaller angular momentum relative to its energy.
Beware of placing too much importance on the central object, though. In the case of, say, an accretion disc around a compact object (white dwarf, neutron star or black hole), the central object totally dominates the behaviour of the orbiting material. In the Solar System, the central object (the Sun) mostly dominates the orbital behaviour but clearly there are smaller systems where other objects rule, like planets over their moons. In the Milky Way, the central black hole actually only dominates over a small region in the centre. Our orbit is determined by the black hole and all the stars, gas and dark matter inside our orbit. It doesn't affect the description above but I thought it was worth saying.
It'd be really great if there was an animation of the "smearing out" of a sphericalish cloud into an accretion disc but I couldn't find one...
Well, first of all, the entire site dedicated to the 2012 nonsense is a total hoax... I suggest that you check out this site for more information regarding the weakness and outright lies of that hoax.
To address the copy/pasted nonsense... The charlatans at the site you reference have taken real terms, and mixed them up in a word salad as to make any lies or fantastic tales they tell seem plausible.
For instance, the Sagittarius Dwarf Galaxy is indeed a real thing (although its discovery wasn't specifically tied to dark matter). The dwarf galaxies that are around the milky way are not going to cause any particularly disturbing collisions in the near future. Most of them just pass through the milky way on their regular orbits. The most significant collision will take place in about 3 billion years when the Andromeda galaxy and our galaxy collide. However, when galaxies collide, it's really just a gravitational interaction. Very few (if any) actual stars hit each other).
Also, the solar system is part of the Milky Way, and from everything we know about it, it has always been part of the milky way. It may get ejected in 3 billion years, but until then it shall remain part of the milky way.
The second paragraph you quoted is total nonsense (above and beyond the regular nonsense of that entire site).
Best Answer
The Sun is approximately in the plane of our Galaxy - see this Astronomy SE question. The ecliptic plane (plane of the solar system) and the Galactic plane (the plane of the disc of the Milky Way) are inclined to each other at an angle of 60.2 degrees.
This is a point you can confirm yourself by noting that the Milky Way does not follow the signs of the zodiac (which follow the ecliptic plane).
There is really no reason that there should be any alignment. Star formation is a turbulent, chaotic process. The evidence so far is that this leaves the angular momentum vectors of individual stars, their discs and ultimately their planetary systems, essentially randomised.
The question only asks "What angle does our Solar System's plane (or, normal to plane) make with The Milky Way's plane" -- to which 60 degrees is the answer. To completely specify the relative geometry of the planes we can ask what are the Galactic coordinates of the ecliptic north pole?
The ecliptic north pole (the pole of the Earth's orbit and the direction in which a normal to the ecliptic plane points) is currently at around RA$=18$h, Dec$=+67$ degrees in the constellation of Draco. In Galactic coordinates this is $l=97$ degrees, $b=+30$ degrees, compared with the normal to the Milky Way plane which is at $b=+90$ degrees (where $l=0$ points towards the Galactic centre and $b=0$ is roughly defines the plane of the Milky Way).
See also https://astronomy.stackexchange.com/questions/28071/in-which-direction-does-the-ecliptic-plane-make-an-angle-of-63-degrees-with-gala
Edit: Note that as pointed out in comments, the ecliptic plane is not quite the same as the plane of the solar system. They differ by about 1.5 degrees.