The way I see it
“Fuji-san is pretty” I think
“That’s a cat” I think
The とis quoting what you think 
But what about the question mark on the second one?
The translation in the example is:
Do you think that Mt. Fuji is beautiful?
I still don’t understand that one.
Oop sorry I just woke up didn’t see the ? (´(ｪ)｀）
In that case I would say
Do you think that Mt. Fuji is beautiful? (This that is a quotation helper too isn’t it? )
Do you think “Mt. Fuji is beautiful”?
“Mt. Fuji is beautiful” you think?
(Obviously in normal translations I wouldn’t write it this way, but to help you understand)
Again と is just quoting thoughts here. You understand the questions just fine✨ Just know that , と comes before おもう a majority the time. For me, I didn’t think about how the とwould make sense in English, just that I needed it before using おもう to make sense in Japanese. がんばって it’ll get easier as you learn more and grow
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ありがとう I hope it will and it is good to know that it is common with おもう
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