According to this HiNative Q&A, 「行ってまいります」 is 「少し古い」 (old fashioned) but also 「品のある言い方」 (a refined way of speaking).
Since the girl who says it in the comic panel always speaks in a polite manner, the mangaka probably chose this wording to reflect that. In the anime adaptation, however, her line was changed to 「行ってきます」.
Here I am, desperately trying to keep up (I’m so busy trying to catch up with Yotsuba too that I have no time at all!) and I have to just thank everyone for their questions and thank @Micki and @ChristopherFritz and others for your replies. Christopher, especially, your explanations are amazing! My book is full of notes taken from your answers! It’s like a master-class. You ought to be doing this for a living! Thank you!
I’m looking all over but can’t find it. Can you let us know the page number please?
Ah @Marcus it was in reference to one of the manga examples Christopher posted to explain the -てきます form a few posts back. That’s why it’s a bit off-topic; it’s from a different manga and not this one. ^^
@ChristopherFritz Thank you, btw, for indulging me! Makes total sense <3
Ah, I see! Nice one, thanks for explaining! I thought I’d gone blind or something! lol!
It might be helpful to edit your original post to make that clear. I also spent five minutes hunting through this week’s pages in レンタルおにいちゃん trying to find it!
I’m managing to keep up with the book and enjoying it, despite being away a few days last week and needing to catch up late (a lot more dialogue in week 3 than the previous weeks with many new grammar concepts for me).
Page 36 bottom panel
叶実ちゃんって書けることあるのかなー
I get the gist of what’s being said here - a taunt. (btw anyone think it’s pretty harsh for the teacher to set an essay on “Recent family memories”? 
) I’m not quite clear on the grammar though, particularly with, I assume, a lot of dropped particles. Could someone please break it down?
As far as I understand this is:
叶実ちゃん => Kanami-chan
って => In informal contexts, this can mean “as for”; it’s kind of like an informal は particle. (BunPro link). => As for Kanami-chan
書ける => The potential form of 書く (can write).
こと => Thing => As for Kanami-chan, a thing (she) can write.
ある => Exists => As for Kanami-chan, a thing she can write exists.
の => Not very sure about this one, but I feel it’s an explication-tone の (BunPro link, Tae-Kim link). It makes sense since the sentence is implicitly asking a question (and thus, looking for an explanation).
かな => I wonder.
“As for Kanami-chan, I wonder if there’s something she can write (for the homework)”.
Wow I didn’t even catch the taunt in there by the teacher. That’s so horrible!
Alongside @2OC3aOdKgwSGlxfz’s breakdown, I (unintentionally) hid a similar breakdown on a prior reply.
When I’m responding to an e-mail that covers multiple topics, after finishing my reply on one topic, one way I’ll switch to another topic is to write:
Now, speaking as to whether cats can dance…
This “speaking of” is similar in sound/meaning to the quote-marking 「って」.
Looked at this way, one can see why it’s said to be like an informal は particle. Just as は formally marks the topic, って works like “speaking of” as another way to switch the conversation over to a different topic.
It was a student who said that, so no bullying by the teacher there. (The side-comment on the teacher was referring to the essay topic.) Thankfully, the classroom isn’t that unfriendly a place for Kanami!
Page 40.
紗絵ちゃんよかったね
I’ve read this as “Sae-chan was good, right?” but how come Kanami is complimenting Sae-chan here? From context I think she’s trying to be the bigger person, but it sounds kinda ood?
何よつまんない
Little confused here too. Is Sae-chan just sulking because Kanami didn’t react the way she wanted her to?
I was confused too by the first sentence but then I interpreted it more like “Sae-chan, it was nice talking to you” (but I’m leaving, therefore, a way of saying goodbye, よかったね as a marker of end of the conversation?). I just thought she didn’t feel that way but tried to keep it cool/non-confrontational with Sae-chan.
よかった, in this context means “good for you”. She’s just saying that she’s thinks it’s a nice thing that Sae’s mother bought it for her.
つまんない means “it’s boring”. I agree that this refers to the fact that Kanami didn’t react like she wanted (and thus, that the teasing she wanted to do ended up being boring for her).
Yes @ChristopherFritz is right, it’s the student in the next panel who’s taunting her. Just above the speech bubble there’s a little sound effect ヒソ… like a pssst or a whisper. It would definitely be too cruel if it was the teacher!
I am so sorry to have caused confusion. I went back and edited my post. I keep forgetting how to quote the original posts, too, so I have to constantly edit.
Page 37 Middle Left Panel
だって参観に来てくれる家族もいないのにさあ
I’m a little confused as to what these two characters are doing/bring to the sentence. I think the sentence itself is saying something like “That’s because her family won’t come to the visit.” and the さあ is just a kind of sassy ending I think. The も is relating back to the panel before it and the いない is negating the action of coming to the vist.
のに can be used for lots of things but I think in this case it’s like “in order to.” In the previous sentence Sae said 書けることなんてあるわけないじゃん (something like there’s nothing for Kanami to write ([for the recent family memory essay]). Then in this sentence she’s saying something like after all, there isn’t even family to come visit. In order for Kanami to write the essay she’d need to have family, and she doesn’t even have some that will visit the school
Here’s a link to the grammar point on bunpro のに - Japanese Grammar Explained | Bunpro
(and hopefully somebody else comes along to explain it better)
だって参観に来てくれる家族もいないのにさあ
I broke it down like this:
だって = because, after all
参観 = visit
に = particle for “to” or “in”
来てくれる家族 = family that is able to come
もいない = does not exist (negative of いる), the も is used as an emphasis
のに = I think technically this means “although” or “in spite of”, but I learned this as a regretful “if only this was not so” when it’s at the end of a statement
さあ = not sure if there is a direct translation, other than “well!” or “come on!”, but I think at the end of a sentence it’s a playful way of joking and/or softening the statement even further.
That’s how I arrived at:
After all, she doesn’t even have a family that can visit.
And the tone I read it in was that of being cruel teasing/joking about someone behind their back, with the さあ kind of being sassy, as you said. 
Thank you!
Thanks!
Hello everyone, I have a couple of questions for this week. Although I feel like they are very basic grammar, I was not comfortable enough to put that away on my “grammar drawer”, so here I go:
Page 37
叶実ちゃんの嘘なんじゃない？
How does that なんじゃない came to be?
From the context I translated as You think Kanami is lying?, but I’m struggling a bit with the grammar here since when I break it down I get Kanami’s lie なんじゃない？, and that doesn’t make any sense.
Page 39
ママは甘いからなんでも買ってくれてねー
Why is くれて in the て-form here?
I suppose this comes from the grammar structure てくれる = “[do] to me/someone near me”, so I translated this as Because my mother is so sweet, she buys me anything I ask her to., but I did not get why くれる would be in the て-form here.
Page 43
紗絵ちゃんのママきれー！
What is the exact meaning of きれ here?
DeepL gave it to me Sae’s mom is beautiful!, but I could not find a correspondent meaning on Jisho/ichi.moe.
Page 44
ごめん遅くなって
Again another problem with the て-form for me: why?
I translated this as Sorry I’m late., but did not understand why the て-form was used in this particular sentence.
Whew, this week was heavy, lots of sentences and new grammar, but it’s fun!! I’m lagging behind because life got in the way (as it often does), but I’m not ready to give up yet. See you all on the next week!