Translation:Sofia finds her puppy very amusing.
It's neither, for these reasons:
Not amuzan: If you would say "ŝi trovas sian hundidon tre amuzan" it would be translated into "she finds her very amusing puppy".
Not amuze: If you would say "ŝi trovas sian hundidon tre amuze" it would be translated into "she very amusingly finds her puppy".
It's therefor "amuza". I know it might seem incorrect at a first glance, but hopefully this makes sense.
I know this is a bit late - but I think the better way to think of this is as having elided a word. That is, I think this sentence is short for "ŝi trovas sian hundidon esti tre amuza." The case is thus not an object, because the verb 'esti' doesn't take objects.<h1>Maybe you already understand this at this point, but I thought this explanation might help if somebody else has this same question.</h1>
Actually, "pup" is the norm for "hundido" in New Zealand, and (here) pups are only referred to as "puppies", by some little kids, and when talking to those little kids that say "puppies", (it's seen as something that kids grow out of).
While the use of "pups" (and "cubs") with the meaning of children is typically understood without problem, actual use seems to be restricted to the set phrase, "What a mucky pup you are", referring to little kids that have habit of getting dirty.
And, (except perhaps in a metaphysical and purely good natured way) I don't really think that referring Sofia's children (or hundidoj for that matter) as "pups" would be interpreted as implying that Sofia is a "Mother dog", (although the hundido might disagree :P)
Personaly I think that "to find" in this meaning, is an archaic phrase from Indo-European languages… But it's logical (so we can use it in esperanto) because may mean "to find sth (in the world) (and recognize it or accept in own mind as it is (in the world)). Twisting but (perhaps) logical :)