1- Here the only reason is that this is a phrasal verb. You can find grammar rules regarding phrasal verbs but I would consider them more like vocabulary, so you have to learn them eventually.
2- The reason why you have to put <em>to </em>instead of <em>for </em>is because it's a dependent preposition. Dependent prepositions are the ones which belong to verbs, adjectives, or nouns. They don’t form a new phrase, and they are often not translated into other languages or used differently in them. You can find lists in Google. In my opinion, you have to approach them the same way you do with phrasal verbs.
3- "<em>try as I might</em>" Is a set phrase, an expression. It means that you are making a very great effort but still cannot do something.
4- Here because the sentence is negative, the question tag is positive:
No one wants to have homework, do they?
Even if the verb is positive, the meaning is that people DON'T WANT homework.
If the sentence was positive, the question tag would be negative:
They want to have homework, don't they?
The meaning of the sentence is clear, they DO WANT homework, so the question tag is negative.
Usually if the main clause is positive, the question tag is negative, and if the main clause is negative, it's positive.
5- Unless means '<em>except if'</em>, but you can also think of it as the negative form of 'if'. So unless = if not. You can rephrase the sentence like this:
You don't have to apply for a student visa if you are not from outside the EU.
You can use <em>if </em>too, you'll just have to change the negative verb into positive:
You have to apply for a student visa if you are from outside the EU.
6- Every year = routine, so we use Present Simple. The Present Perfect Simple is only used with adverbs like <em>already</em>, <em>still</em>, <em>yet</em>, etc. Also, <em style="color: rgb(0, 71, 178);">to have/get</em><em> </em><em style="color: rgb(178, 107, 0);">something </em><em style="color: rgb(0, 97, 0);">done</em> is a causative form:
I repaint my house.
If I paid someone to repaint it:
I had my house repainted.
Hope it was helpful! Good luck with the exam :)