As it takes the sun about 250 million years (250 My) to orbit the galaxy, the proper motion of stars relative to the Sun will be the dominant effect of changes in the sky. The visible effects of the rotation will be far slower.
All stars move in the sky, some faster some slower, and in more or less random directions, not just moving around the galaxy. For example, Vega moves about 1 degree every 11,000 years. Around 12,000 BCE it was the pole star, and will be so again around 14,000 CE. Between now and then, other stars like Gamma Cephei and Iota Cephei will temporarily take the role of Polaris.
By 250 million years most stars will be far from their current position in the sky, but because of uncertainties it's impossible to say just where they'll be. For example, if the estimate of 250 My is wrong by just 1% (or 2.5 My), that means about 100 periods of Vega. Hence by that time Vega could be anywhere at all even if it stays in our general neighbourhood - which is certainly not guaranteed.
Galaxies move as well, but because they are much further away, their apparent position changes much slower than that of stars. It will be mainly our rotation around the galaxy that moves them in the sky.
Using telescopes we have already seen differences in the positions of the closer stars. In 10,000 years many changes will be visible to the naked eye. By the year 250 My, the sky won't look even remotely like the present.
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 Wikipedia article on "Stellar density" says the stellar density near the Sun is only 0.14 stars per cubic parsec. It suggests that the density in the central core and in globular clusters is about 500 times as great.
According to the List of nearest stars and brown dwarfs in Wikipedia , there are 61 stars within 15 light years of the Earth. Dividing 61 by the volume of sphere of this radius, we obtain 4.3e-3 stars per cubic light-year, or 0.15 stars per cubic parsec. Thanks to user31264 for providing this information, which is consistent with the information from the previous link.
According to the Wikipedia article on "apparent magnitude", the total integrated magnitude of the night sky as seen from Earth is -6.5. Making that 500 times as bright produces a total magnitude of about -13.2 (5 magnitudes is a factor of 100 in brightness). The maximum brightness of the full Moon is -12.92.
So even with 500 times as many stars in the sky, the total brightness would be only slightly greater than that of a full moon.
(This assumes that the average brightness of the core stars is similar to the average brightness out here in the Galactic suburbs.)
Parts of the core might be even denser than that.
(I've updated this with new information and deleted and old link whose numbers appear to have been incorrect.)