http://www.nasa.gov/worldbook/artificial_satellites_worldbook.htmlCan't read it all right now (I will later though; :) I <3 learning), but I assume it's all top-tier info; straight from NASA.
My guess for your questions..
If an object were to be "pushed" from somewhere in space to an area close enough to Mars that it would be pulled in by it's gravitational pull, it would orbit Mars. Similar to how we have constantly orbiting space stations (and the moon, of course). Those objects have to be going at a certain speed to stay in orbit. If they go too slow, then they will be pulled down (towards the center of the earth). If they go too fast, they will be launched into space. Ever played with gravity? Hold a quarter on it's edge by your finger tip and a flat surface. Flick the edge of the quarter. It will stay relatively straight up and down (if you were to draw a straight line in the middle of it and the two points of the line were touching both you and the flat surface when you flick it) for a little bit, but as gravity pushes it down, it starts to wobble and eventually falls on one of it's two faces (heads or tails). This is similar to how satellites work, from what makes sense to me (I've never researched this; it just makes sense to me).
Think of when you first flick the quarter. It will wobble and move around a bit. Something similar to this would happen (I assume) if it were to be added into the pull of something like a planet. It would orbit in an elliptic fashion rather than circular, and it would be quite irregular. Hard for me to explain in words, but kinda like it would start to sling shot and then be pulled back in at an odd angle and speed, unless it were to go too fast. After it starts to level out (the object would be at a constant, "perfect" (can't literally be
perfect, but it is valid conditions to stay in orbit) speed and orbit), it is at the ideal speed to orbit. Eventually the quarter slows down and starts to become parallel to the flat surface, ultimately laying flat with no movement. This shows the object becoming too slow and then crashing onto the planet.
Hope that helps and makes sense. :) Dunno if anything I said is right, as I have done absolutely no research or asked anyone about it. It makes sense to me, so yeah.