What would happen if a rocket traveling at speed of light would collide with a planet (like earth)?
Let's assume it is possible for a spacecraft to travel at the speed of light (I've read the interstellar book by Kip Thorne, apparently this is theoretically possible if you swing around two black holes)
I have watched the video where the traveling of a light particle is simulated and while planets are certainly big enough so you can adjust your route, what would happen if a rocket (or spaceship in general) would collide with an earth-like planet (i.e. a planet having a massive surface). Would we burst through the whole planet, make it only to a core, or just a few km? I know what happens when a meteor lands, but obviously they aren't traveling at the speed of light.
And one similar another question - when a spaceship is traveling at the speed of light, is there a problem with collision with smaller objects? I think you can't avoid millions of rockets in outer space, it would surely damage the spaceship, wouldn't it?
It is not possible to travel at the speed of light. It would require an infinite amount of energy. Please rephrase your question to say "99%" of the speed of light, or 99.9% or some other real value.
I don't know about a spaceship, but the XKCD guy wrote an interesting article about what would happen if Earth was hit by a solid asteroid, travelling at various different speeds:
what-if question: Diamond
The ship would be a lot smaller than an asteroid, so I think the damage would be a lesser version of these descriptions. (The most extreme case described is the whole planet being vapourised and even neighbouring planets being affected by the radiation.)
As for the second part of the question - protecting the craft from minor collisions - I admit I have no idea how to work out the kinetic energy from collisions at such high speeds. But even a grain of sand would at least damage the front of the craft, and I guess some sort of thick shield would be needed to protect against erosion from interstellar dust...
Even at normal orbital speeds, Space Shuttle windows have had minor damage as this paper describes...
At near the speed of light there would be huge problems with collisions to overcome. Not just the high heat of impact and likely fusion of what you crash into at high speed, but at close enough to the speed of light, particles will actually pass through each other (the what-if article touches on that). A google search will provide a number articles on the difficulties with near the speed of light travel. Here's one I like: http://www.space.com/8011-warp-speed-kill.html
Here's another one (short and sweet) that touches on tiny sand grains. http://www.csicop.org/si/show/on_problems_with_near-light-speed_travel The good news is, there aren't many sand grains in space (er, I think), at least until you approached other star's Oort Clouds.