This is a big one, so I'll split it into parts based on the questions you asked:
1) Earth has roughly a distance of 1 AU to the sun, Saturn between 9 and 10. Sun's gravity would therefore have a much stronger influence on these rings. Would it already be enough to distort the rings (e.g. even leading to the sun "stealing" parts of it) or would the rings still have a ring shape?
Sun would not have too much of an influence because the rings are all individually gravitationally bound to the earth (similar to the moon, and you don't see moon being affected by the sun, gravitationally; the rings will be much closer to the earth and hence even less affected).
The moon, on the other hand, would distort a few rings due to orbital resonances (similar to Titan Ringlets: https://en.wikipedia.org/wiki/Rings_of_Saturn#Colombo_Gap_and_Titan_Ringlet and resonant gaps in asteroid belts due to Jovian planets: https://en.wikipedia.org/wiki/Kirkwood_gap), and its effect would be much more pronounced because it is closer to the earth than Titan is to Saturn, though limited to a few resonant regions. There will also be ripples if the plane of the moon is different from the plane of the rings (http://blogs.discovermagazine.com/badastronomy/2009/06/11/saturns-rings-do-the-wave/#.U7VjpvmSw08).
2) How well would the objects in such rings be visible from earth? The
Saturn rings are mostly dust and ice as far as I know, but there are
surely some larger objects. Depending on material and size some object
would probably be visible to the naked eye, right? Is there (probably
yes) a formula to calculate what materials and sizes are required to
achieve this?
Most matter in Saturn's ring is particles up to a few metres in size (http://saturn.jpl.nasa.gov/faq/FAQSaturn/#q6). Resolution of the human eye is about 1' (Is the moon only 60 pixels?), which means at a distance of around 1 to 1.5 Earth radii (given by Roche limit of the earth; this is where the rings will be), the eye can resolve things above a couple of kilometres in size. However, Saturn also has Shepherd moons, which are a few tens of kilometers in size. Such moons, if present, would be visible to naked eye. Also, the gaps would be visible clearly, since they are often big enough.
3) Would those rings not be constantly raining particles down on
earth? Would there be a much higher risk of dangerous impacts for the
earth's population?
There would be collisions/interactions in the rings (I'm guessing, correct me if I'm wrong) which would send particles earthwards. However, the particles which will get thrown will be the lighter, smaller ones (conservation of momentum etc.), and usually small meteoroids are fairly harmless. They will also be slower than the usual meteoroids that enter the earth at above 20 km/s, since the orbital motion of the earth doesn't add up in this case. I am a little unsure about the consequences of them being slower, but anyway, smaller meteoroids should technically be fairly harmless. (Also I couldn't find decent references other than: http://science.howstuffworks.com/question486.htm)
4) Are the colours used in the video precise? Don't they depend on the
angle? What influence would our atmosphere have onto them?
I believe so. This section shows true colour images of the rings from two different angles, and there doesn't seem to be a significant change in the colouration: https://en.wikipedia.org/wiki/Rings_of_Saturn#Subdivisions_and_structures_within_the_rings. The atmosphere will probably affect the rings closer to the horizon, with redder colouration similar to sunrise/sunset and moonrise/moonset.
5) Would those rings have the same orbital inclination or would theirs
differ from earth's to an extent where the difference becomes visible?
Depends largely on the formation history. This FAQ question here discusses this fairly well: http://saturn.jpl.nasa.gov/faq/FAQSaturn/#q14
6) What shadow would these rings cause on earth? Would they be dense
enough to "shade out" complete stretchs of land?
There will be shadow similar to that due to clouds, since the rings are not too thick or dense. In many of the Cassini photos, you can see through the rings, so it is likely that the sunlight would still pass through, but would be dimmed (it would be interesting to see by what factor; I cannot say for sure with my knowledge). Also, remember that the earth will still be rotating but the ring system will be dynamically static, so there will be extended periods of darkness during the day for some regions at different times of the year.
I will add more to this if something comes to mind. I hope this was helpful.
