What's the range of my Ferret?
Solution 1:
It feels like a little less than a screen-width. However, I've found them to be unreliable. Sometimes they'll ignore gold that fairly close until I re-summon them, at which point they'll notice it.
No. Regardless of your pickup-range, ferrets have to stand on the gold to pick it up.
Yes, and there's no way to stop them. There's been a good number of people requesting a passive mode, so hopefully it'll appear at some point.
Solution 2:
Yes, the ferrets attack enemies. A friend was commenting on how it's really annoying that they do this, because they're apparently really slow. By the time the ferrets get to picking up the gold, he's already moving forward and has to wait for the ferrets to come back to him. I'm not sure how much damage they do compared to other companions, but I would imagine that it's less (pure speculation).
I'm not sure about their range, or how it's affected by your stats.