As I have not yet finished reading for this week, I dare only comment on your last question:
聞ける, according to my dictionary, is either the transitive version of 聞く and means „to tell“, or the potential of 聞く and means „can tell“. For the ほど construction, I find the potential version more suitable, like you said: „to the extent that I could not speak“ or in better English: „I could not even speak“.
(Does this answer your question?)
Yes I think it does mean characteristic, but it is not Icchan who is delighted, he is saying that Picotto-obaasan has the characteristic of appearing delighted when she tells someone off.
He’s suggesting that they should deprive Picottobaachan of that pleasure.
Oh my goodness, I thought that whole conversation was about how they just find it funny now when she tells them off. Talk about missing the point entirely!
Well thank you @Kyasurin 
Disclaimer first, it’s been too long since I read this book, so I may be way off here. 
I believe it is ピコっとばあさん that seems joyful when she is scolding someone.
From the context of your quotes (I hope there’s not too much happening in between haha), I believe he is suggesting they take away ピコットばあさん’s fun at scolding people - that is, not give her a reason to scold. The しまう in this case would be ‘to do completely’.
がのこる is a pretty common phrase in Japanese books in general, I’ve found, when describing a changed state. If there’s more to it I’ve never bothered to find out, so if someone knows… 
Annd, the last one has been answered sufficiently, I believe?
どういたしまして
It’s good to know we agree on matters grammatical 
By the way, does it bother anybody else that Rina is drawn with “blonde” hair in the illustration on page 117? Her hair is always shaded in every other illustration… For a second I thought the chapter was going to be about some other kid who steals her umbrella 
I’ve given up on understanding the pictures in this book. My copy has the lady entering on the elephant looking calm and even happy.
Yes! That also annoyed me greatly when I read that part. She looked like some kind of zen circus performer, not a distraught, wet and bedraggled queen.
The illustration from the end of this chapter is also misleading / inaccurate 
it’s a shame, cos they’re cute illustrations.
ところがピコットばあさんときたら、居間のまどからはさるすべりが見え、自分の散歩する家の西がわにはつつじがさき、うら庭にはすいせんが黄色のじゅうたんみたいにさいていないと気がすまない。
Is the bold part the masu stem of 咲く? If yes, why? How come it’s not te-form? What’s the difference?
ますstem has the same function as て-form, it’s just more formal and usually used in writing (very often in books 
). E.g. 見え before the previous comma is also in ますstem form.
だって、ぼそっとしゃべって、すうっと消えるんだもの。ぶつかろうにもね……。わたし、大声でどなってもらったほうがまだいいみたい。
Edit: 3) What would be the difference between
子供に会えないということがどれだけ辛いかあなたにはわかりません。
and, for instance,
子供に会えないはどれだけ辛いかあなたにはわかりません？
Edit 2: 4) 「ナータからきいたわ。」「リナは、信じられますか。」
信じられますか = “Can you believe it,” right? I keep getting confused by “potential or passive.”
Edit 3: 5) 眼科にとびこむ
“Jump into opthalmology”?
6, I guess) Did Shikka just say there’s some prophecy about a girl coming to their town?
こしかけさせさせ = 腰掛ける + 差す’s last meaning: “to stop in the midst of, to leave undone,” right?
58% (page 118):
“But, he just talked in a whisper and then immediately disappeared!”
Yes, I think that’s right.
59% (page 121):
Yes, I think it’s potential. If it were passive he would be asking if Rina could be believed, which would be odd in the context.
I took this to be shorthand for “ophthalmologist”, in the sense of “I would rush to the ophthalmologist’s” (because she wouldn’t believe her eyes). I could be wrong on that front though.
60% (page 121)
61% (page 122)
こしかける is correct, but I think this is させる, “to allow someone to do; to make someone do”. I thought he was offering her a comfortable chair / making her sit down.
Oops, sorry 
I did see the bold part initially, and then just lost my head in between that and answering…
I found the meaning “it’s not possible” for にも, which it says is used after the volitional (or dictionary) form, so I’m pretty sure that’s the construction here, because ぶつかる is in volitional form.
So I think she’s saying it’s just not possible to ぶつかる him, because of what she said in the previous sentence - he vanishes almost before he’s appeared.
Well now we’re both uncertain at least 
Yup, the にも fits. I couldn’t find it on BunPro, but I found it in The Handbook…, in case you or anyone else is interested:
問題のつばかさらかがあるというのですね。
I don’t get the bold part.
It’s 問題 の つぼ か さら か
I kept being tripped up by this too because of all the kana. It’s saying “the pot or plate in question”. The nouns are marked by か because of the uncertainty over whether it’s a pot or a plate.
Edit for clarity: this is from page 126, which is week 11
