There isn't a one-to-one correspondence between その and English words. Often it is best translated "that" and would be awkward or incorrect to translate it as "the", but there are also circumstances in which it is much more natural to translate it as "the" and where "that" feels less natural to me.
In an exercises like this there's not really enough context to know, so I think both are totally valid.
There is a big difference. Though they both mean the same thing when translated to English (I prefer translating it to 'that' because of how K-S-A-D words work in Japanese), they are used differently. As you can see from pariah_ls example, それ never has an noun following it, while その has to have a noun after it. You could think of それ as 'this one' and その as 'this X', replacing the X with the following noun. それ is actually a noun itself, while その is an adjective.
I hope this clears it up a bit.
It could be a complete sentence in English, given context.
Like "What are you looking at?" "That menu."
Or "What did you need?" "That menu."
Although yes, I do think it'd be weird to say this without prior reference to the menu unless you use the sentence to urgently point it out for whatever reason you may need to.
they are two different kanas, に・ニ and にゅ・ニュ。they are called "Combo kana" or "Yōon" on kana charts. They are basically a representation of a contraction of a sound.
Here is a link for 拗音【ようおん】:
If you want to type it in a rōmaji input you can type xyu, lyu for ュ、or just type "nyu" and you will get ニュ