Second question, just out of curiosity, would it be possible to look at closer objects like Mars or the moon ? We'd have a pretty good image too ?
Mars in comparison is minuscule in our night sky. It's just a tiny blob of light.
Correct, although wikipedia says 6 times and adds a clarification:
"Although it appears more than six times as wide as the full Moon when photographed through a larger telescope, only the brighter central region is visible to the naked eye or when viewed using binoculars or a small telescope." [1]
Earth - Hubble is moving too fast to get decent close-up imagery, the distance is too small, and frankly Hubble just wasn't designed for it (as opposed to spy satellites of the major powers). Hubble does use the earth to calibrate its cameras, though.
Randall gives examples of how imagery might turn out.
Moon - speed's an issue, but less so, especially if you're imaging a large region
Here's an example: http://hubblesite.org/newscenter/archive/releases/2012/22/
Mars - Hubble has targeted this planet lots of times.
Some examples: http://hubblesite.org/gallery/album/solar_system/mars/
Maybe you didn't zoom far enough? At sub-maximal zoom, there is aliasing resulting from a lack of low pass filtering before the resolution reduction. That looks a bit like sensor noise.