I think maybe she is saying ておくin both cases and just dropping a sound, and that the standard form would be 洗っておいてあげる and だしておきなさい。
And now I finished reading through I realise this was also answered above. 
I really need to read everything before I get excited about answering questions.
Thank you all for your help! I really appreciate it. Please look forward to my next set of questions 
@Kyasurin - nothing wrong with being excited about answering questions!
A question and clarification of the ending scene. Thanks as always for your help!
あそこをまがると、いつもの古本屋のはずなんだけどな。こんな店あったかな。
When turning the pages of the book,(this is the definition of まがる in the sheet), it seems to just be a usual second hand book store. I wonder if there’s a shop like this.
At the end, Natta says that the town is always a step away, and that they came to great Rina. Rina says this isn’t true, but it seems like Natta is hiding something from her. Then a yellow duck flies away saying mazaaguusu…
I suspect somebody added this after the later めくる, and didn’t notice excel autofill “the pages of the book”. まがる just means “to turn”, and I understood that he was talking about turning a corner (in some street wherever he came from), and coming across this place.
This was one of the pages / scenes I had questions about, so I don’t want to speculate on the rest of it.
It’s saying “Mother Goose”
The katakana tripping me up again! 
But I genuinely think はいる is just about one of the worst words to be in kana. The fact that はいって could just as easily be a particle plus a different movement verb (thus also usually fitting the context) is a killer combination.
I have more questions plx kind folks 
Page 72
イッちゃんが、赤い鼻にしわをたくさんよせて、むずかしい顔で、あんまりまじめに残念がるので、リナは、飲みこむのに苦労した。
Okay, I want to check:
Page 77
めがねのおくでわらう - is this, like, “she laughed from behind her glasses”, sort of thing?
四日めともなると、- I’m a bit list with ともなると, though I’m guessing the meaning is “on it becoming the fourth day”?
この暑いのに予備校がよいじゃたいへんね。
… Okay, first of all I’m confused by 暑いのに. Is it の and then に separately (-> “in this heat”), rather than it being the specific のに particle?
And then what’s with よいじゃたいへん? I kind of get that it’s saying it’s “not good”, but it seems supremely weird to that something’s よい is たいへん, as it were?!
Page 79
I am not gonna lie, it took me so long to realise the “poetry anthology” is “Mother Goose” - I thought somebody just randomly brought up Mother Goose while this kid was trying to buy a book of poetry - and even so I still don’t understand what’s going on.
「つぎ、マザーグース。大きなあひるの絵の表紙だから、すぐわかってよ。」
What??? Who is even talking here?! And… “it has a big duck on the front, so you’ll understand soon”???
そんな人にしか本を売らないよ。」
So, I got quite confused here. I initially thought it meant “I only sell books to that person”, but that seemed way too absurd. Then I realised it was そんな - she’s actually saying “I only sell books to such people”… right?!
So… does she just not charge? She just lives off of other people’s joy in books? Je suis confused.
Rina’s followup comment is あれでも売ったっていえるの？ - I’m not totally comfortable with あれでも?
I’m not sure I can help with your questions in pages 72, 77, but maybe I can clarify 79.
My understanding is that here ナータ is talking to Rina. They are busy working sorting the sea (now lake) of books, so she’s just telling her what book she has too look for next (a book with a big goose in the cover, so she should be able to know which one is it as soon as she sees it). Just before this part it says:
そういって、だれもいなかったように
This refers to how ナータ considers the business with the boy complete and is just returning to what they were doing before as if he wasn’t there anymore.
My understanding is the same as yours. しか is often translated as “only” but I believe it is always a lot less confusing if you use “except for”, because the verb is in negative. She’s saying that Except for that kind of people, she doesn’t sell (to anyone).
From where I see it, it doesn’t quite say that “she lives off it”, but more like she feels the actual compensation / satisfaction comes from the fact that her customers really like those books (and thus that she has found a good owner for the book, I guess?)
I am also not good at finding a good english equivalent for あれでも, but in this particular case I feel this uses the “even” meaning of でも, and “that” for あれ. Can that even be called “selling”? would be my best attempt at somewhat natural English for that sentence.
Oh, so they are talking about something else? This actually clear up my confusion from page 80 as well, so you’ve even fended off a future question I got that either ナータ or the boy were suddenly talking as if nobody was there, but the actual comment just seemed so out of left field that I didn’t understand what they were talking about 
Oh yes, this is how I always think of it 
I was just confused by the そんな人 part.
Yes, sorry, I do agree with you - it just seems pretty incredible that she seemingly never charges
Ah, thank you - for some reason I thought she was referring to the boy (I don’t know why, that would be quite rude!), but that makes total sense.
Itcchan says he will invent a medicine to take away tiredness. John says he invented such a thing ages ago (ie his cooking revives you). Rina understands what he is saying (she’s really enjoying the steak) but Itcchan misunderstands, and asks if John has already been granted a patent (for his invention). And he says it with such disappointment that his face wrinkles up and Rina has to try really hard not to laugh out loud.
Ah, thank you - I did understand all the context up to that point, but I wasn’t quite certain of Rina’s reaction - I thought it was more that she thought it was hilarious so didn’t say anything, but even so felt bad for him - but your explanation makes much more sense.
Also, just realised I should have searched しわをよせる directly - I was thrown because none of the definitions of よせる quite seemed to fit.
めがねのおくでわらう - is this, like, “she laughed from behind her glasses”, sort of thing?
Yes I think so. Like, you could see it in her eyes rather than hearing her laugh out loud.
四日めともなると、- I’m a bit list with ともなると, though I’m guessing the meaning is “on it becoming the fourth day”?
I think maybe we’d say “by the fourth day” (the sea of books had become a lake of books). It’s putting a も into ~となると
この暑いのに予備校がよいじゃたいへんね。
… Okay, first of all I’m confused by 暑いのに. Is it の and then に separately (-> “in this heat”), rather than it being the specific のに particle?
And then what’s with よいじゃたいへん? I kind of get that it’s saying it’s “not good”, but it seems supremely weird to that something’s よい is たいへん, as it were?!
I think it’s separate. And I think 予備校がよい might mean “the commute to cram school”, from 通う（かよう)。So I think she’s saying it’s たいへん to commute to cram school on a hot day.
I read it as the ‘even though’ particle combination, but I suppose it doesn’t change the meaning much. 
Even though I read it like that I would not put ‘even though’ in the English sentence. Maybe just ‘even’ would carry enough of that nuance… 
I forgot to track this thread. I hope everyone is doing well.
Page 73
型をわるくしてしまってよ。
This bit where Rina says she’ll wash her own clothes and キヌさん tells her she’s too inexperienced and she’ll mess the dress has been discussed already, but what does 型 exactly mean? I’m not particularly happy with any of jisho’s definitions.
Page 76
男の子は、またにいっとわらった。リナは、まあという顔で、ぷっとふくれた。
How is いっとわらう different from a normal laugh/smile and why is Rina pouting at it?
Page 78
まえから読みたかった詩集だ。
I’m not sure about まえから here. I take it as if he’s saying that he’d prefer reading a poetry anthology rather than the maths book he was looking for, but I don’t see how まえ (before) and から (from) could mean that.
One of the definitions given is “size (i.e. clothing, shoes)”. If that’s the right meaning here, it would mean that she would mess up the size of the clothes (i.e. shrink them). I’m not that confident in this explanation though.
It’s にいっと together. As to how it’s different? 
He’s wanted to read that poetry anthology from before. That is, from some unspecified time in the past. This goes along with 読みたかった being in the past tense.
This makes so much sense, but somehow my brain hadn’t thought of it this way. Thanks!
形を悪くする
I think it could be something like to ruin the shape, deform it.
https://www.google.it/amp/s/ejje.weblio.jp/content/amp/Deform
This is actually p76
Weblio defines にいっとわらう as 軽く笑顔をつくること, so I guess it is a slight smile? 
EDIT: Actually I realise the Weblio entry was for 二っと笑う but I don’t know if there is really much of a difference. Jisho says にっと is “with a grin”.
Is Rina is pouting because he’s teasing her about her weight? I seem to recall she worried about that at one point?
I didn’t know 型 and 形 are interchangeable in this situation, but now that you said it I realized that the Japanese dictionary on Mac (which I heard somewhere sometime that is good) lists both kanji for the reading かた and the definitions we were talking about, so it makes sense!
Oops, I was going to write the first appearance of the word which is a couple of pages before but changed my mind. Edited!
So there is an extra い just to make it difficult to look up in a dictionary 
All clear now.