Yes, Kim is a raised object in (2). But it's not in (1) or (3). They all come from similar structures, a main clause I didn't want
NP, where the NP is a complement clause with Kim as subject and mistreat my cat as VP.
- I didn't want [np [s [np Kim] [vp mistreat my cat]]]
but the derivations are independent, largely because of complementizer choice.
To get (2), which uses an infinitive complementizer, Raising is applied, producing a parse like this:
- I didn't want [np Kim] [vp to mistreat my cat]
which separates the raised object from the infinitive clause, reducing it to a phrase.
But Raising doesn't apply to most gerunds, so in (1) and (3) the gerund complement clause is still a clause and it's still the object of want. In (3), Wh-Cleft applies, inserting the what to introduce the construction and the was as fulcrum of cleavage to separate the cloven constituents.
- What I didn't want was [np [s [np Kim] [vp mistreating my cat]]]
Note that, while want does govern B-Raising with infinitives, an NP raised to object position of want does not passivize:
- They wanted Bill to replace the toilet.
- *Bill was wanted to replace the toilet.
even though most Raising verbs do allow this:
- They expected Bill to replace the toilet.
- Bill was expected to replace the toilet.