- It is hard to think of an example where I would expect 思う to take an object, other than when thinking about something e.g. 母のことを思う. I wonder if the を here is the object of 育てる rather than 思う.
It would help you if you could somehow forget the notion "思う = 'to think'" for a moment. I could be wrong but I feel that might be what is preventing you from understanding this.
"To regard A as B" is the expression that I would like you to consider here because, frankly, that is what 「AをBと思う」 means in the given context. Yes, 「思う」 can take 「を」 and 「と」 at the same time.
Which leads me to ask can を act through a て joining two clauses such that it is the object of the second clause?
If you are seeing two clauses in that sentence, it is a proof that you are reading the sentence structure incorrectly.
It is not in the structure of "One does thing A and then he goes on to do thing B." That is not the usage of 「て」 in the sentence in question. The 「て」 here expresses simultaneity of (1) raising the girl and (2) regarding the girl as his own. Your translation says another thing.
- I've never seen 〜と思うcome after anything other than a verb/adjective/copula. Can it act on a noun e.g. 猫と思う (I'm thinking of cats)?
「猫と思う。」 does not mean "I'm thinking of cats." That would be 「猫のことを考えている。」 or 「猫のことを思っている。」.
「猫と思う。」, all by itslef, does not really mean anything. 「猫だと思う。」 means "I think it is a cat." as an answer to the question "What animal do you think that is?"
- If I were asked to translate into Japanese I would probably try something like Bobはこの女の子が自分の子供だと思って育てることにしました。Would this be wrong and, if so, why?
No, it would not be wrong; It actualy sounds pretty natural as well.
- Can someone please break down the original sentence to explain why it is correct?
It is more than correct and grammatical; It is perfectly natural on the native level (except, perhaps, for the silly-looking "Bob" part.).
Bob made a decision. ← 「Bob は～～～～ことにしました。」
What was his decision? It was 「この女の子を自分の子供と思って育てる」.
= "Performing Action 「この女の子を自分の子供と思う」 and Action「（その子を）育てる」 simultaneously."
Put it all together and you should have:
"Bob decided to raise this girl by regarding her as his own (child)."