However, if you have shot a ship (and hit it), and it then dies from non-combat damage soon after (by crashing into an asteroid, for example), you will be treated as the ship that destroyed it.
Dealing with who is to blame when collisions occur is quite difficult, if not impossible to work out. It's not something currently on our sensors to address.
Larger ships will, generally speaking, not bother to avoid smaller vessels in combat. Smaller ships definitely attempt to avoid larger vessels though (we're also updating collision damage to favour ships with greater mass).
You can sometimes force a collision (with effort) with a ship that is actively avoiding you, and occasionally a collision will occur because of either A) a very poor AI pilot skill level or B) a combination of ship positioning in a fur-ball that gives the AI no reasonable manoeuvre options.
In general, the smaller your ship, the more worried you should be about collisions (and of course, the more able you are to avoid them).