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しろ is the imperative/command form of する.
From the linked article:
The command form is a verb conjugation that changes verbs into commands, orders, and demands. As you can imagine, it is pretty strong and not exactly super polite to use.
Rather than saying he’ll call Takeo, he’s telling Takeo to call him.
行け is the command form of 行く, so it’s the same thing. Takeo isn’t included (as he just graduated), so he’s telling the club to go to nationals this year.
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Consider someone asks why you lent money to a person you know won’t pay you back. You could reply with:
- He needed the money.
- It’s that he needed the money.
Taking the main sentence and changing it to “It is that (sentence)” gives an explanatory tone, like you are giving the reason for something rather than just a plain statement.
Or, as a question, it gives the sense of asking for the reason that something is.
- Why are we friends?
- Why is it that we are friends?
I wrote a bit about the grammar with examples in this post, which I highly recommend reading. I have additional examples on this web page.
Here, けっこう has the meaning “quite” (quite a mystery) or “very” (very mysterious).
I think if it had “lovely” as a translation, it would be けっこうな, but I may be mistaken.
けっこう is one you get a feel for over time. It’s a fairly common word, so you’ll undoubtedly see it a lot if you keep up reading native material.
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This manga was published by a company called 集英社. The middle school name came from that. It’s kind of like an Easter egg =D
It often gets translated along the lines of “something like” and typically is used for something disliked.
Rather than saying “Going to a separate high school is lonely” it translates more like “Something like going to a separate high school is lonely” with the added nuance that he dislikes it.
If you’ve learned about よ as emphasis, this is similar to that.
Japanese has various particles that go at the end of a sentence that can add emphasis (よ), or make them into a question (か), or seek/expect agreement (ね).
If よ is like putting ! at the end of a sentence, you can think of ぜ as putting !!! at the end. It’s also more rough/crude, so you’re likely going to see among boys as a group, or from tough guys around anyone.
For this one, I’d lean toward “lonely”.
と can mean “with”, so おまえと is “with おまえ”.
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Yes. 
Yes. We’ll probably be seeing this one a lot!
It’s sort of like how in English you might say “What would you like for dinner?” but it’s verbalized as “What wouldju like for dinner?”
Similarly, “かえるの” becomes “かえんの”.
This would use the completed form of a verb (past tense), but 帰る is non-complete (present/future tense).
帰る = going home
But what if we add の to make “going home” into a noun?
帰るの？
“Is it (that you are) going home?”
We’ll probably see this use of の quite a bit, so don’t hesitate to ask about it if needed in the future. And my post on it that I linked to above will always be around to refer back to!
Here, Takeo is asking the same question back to Suna:
帰るか? = Going home?
In context, this most likely means going home. (Stay tuned for Week 2 to find out where they’re going!)