The dark matter isn't directly responsible for the rotation of the Galaxy - that is a consequence of the initial angular momentum (or the angular momentum it has accrued during its formation).
Dark matter is required to explain the rotation speed as a function of radius that are observed in the Milky Way (and other spiral/disc galaxies). The dark matter in the Milky Way needs to be distributed pseudo-spherically on a scale that is much larger than the distribution of the normal, luminous matter.
It is correct to say that the Sun "orbits" the Galactic centre, but it is the combined effects of all the mass in the Galaxy that define its orbit, not any central object. As a result, the orbit is not "simple" (like the orbits of planets around the Sun) and it wobbles up and down and in and out and it is not "closed" in the sense that it will not follow exactly the same trajectory each time it laps.
Within the 8 kpc galactocentric radius of the Sun's roughly circular trajectory around the Galaxy, it is the normal luminous matter of the bulge and disk that provide most of the centripetal gravitational force on the Sun; dark matter dominates orbits only at galactocentric radii of $>10$ kpc. Without any dark matter we would still expect the solar "orbital" speed to be about 180 km/s, compared with the 220 km/s it actually has (e.g. see https://physics.stackexchange.com/a/169141/43351 ).
The second part of your question conflates spiral structure with "order". The difference between spiral and elliptical galaxies is indeed that most of the stars and gas in spiral galaxies orbit in a plane with approximately circular orbits, whereas stellar orbits in ellipticals have a much wider variety of orbital inclinations and eccentricities. This difference is not really attributable to differences in mass (whether total or just dark matter), it is a consequence of how those galaxies formed and how star formation occurred in the gas that was present.
In a spiral galaxy, there was plenty of available gas that collapsed to a disc prior to the majority of star formation - this leads to the observed "ordered" orbits of the stars today. On the other hand, if a galaxy forms from the merger of essentially gas-free smaller units, then the stars form a collisionless system and the orbits get scrambled during the merger - leading to an elliptical galaxy.
Measuring how much dark matter there may be in elliptical galaxies is considerably harder than for spirals - precisely because they do not have stars and gas on ordered pseudo-circular orbits. However, measurements of masses using strong gravitational lensing do indicate that most ellipticals do contain significant amounts of dark matter within their luminous regions (e.g. Bovy 2023). i.e. The amount of dark matter present is not what explains the differences between spiral and elliptical galaxies and since elliptical galaxies can be as massive as the most massive spiral galaxies, it isn't total mass that explains their differences either.