The way I see it
“Fuji-san is pretty” I think
“That’s a cat” I think
The とis quoting what you think 
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
ありがとう I hope it will and it is good to know that it is common with おもう
Would it make sense to use と in sentences like :
Today I wrote to a friend I made in Japan. I wanted to tell them I learned a new kanji. I phrased it like this:
今日は、「床」の漢字勉強しました。
But during my Bunpro grammar study I just learned about と being used after quotation marks and now I’m questioning if the above is correct or if there’s a better way to say it.
Your sentence is fine, it will get the meaning across.
In this kind of situation you can use this construction: という (JLPT N3) | Bunpro. Just と on its own is not enough.
You can also quote with って (JLPT N5) | Bunpro if you speak more casually.
I’m confused by the first review sentence: " 彼はとても親切だと聞きました。"
だ is part of the necessary response, but it seems to be part of what is being quoted. None of the other review sentences include that. Is it doing something else here that I’m missing?