4
I would say あっちゃ, こっちゃ here in your example are the contracted pronunciations of あっちへ, こっちへ.
うまい もも こっちゃ こい。＜＜ うまい桃、こっちへ来い。
"Delicious peach, come this way."
にがい もも あっちゃ いけ。＜＜ 苦い桃、あっちへ行け。
"Bitter peach, go away."
3
As you said, this で is "at" (at a place). 日之出ん家で means "at Hinode's house". ～ばいいのに has the other meaning, which is a suggestion such as "why don't you ～". This する in the manga means "take a poop". So it means "Why don't you take a poop at (the bathroom of) Hinode's house?"
2
どうにもならない (not どうにかならない) and 仕方がない are similar like you say.
どうにもならない is however reserved for significant things as it conveys irreversibility. E.g. the following is fine:
電気消すの忘れた
仕方がないな
The following is odd because surely it's easy to go back and just switch it off.
電気消すの忘れた
どうにもならないな
The following is thus natural again:
あー、家出る時電気消すの忘れた！もう飛行機あと５分で出るのに
...
2
Here 「こっちゃ」is a diminutive for「こっち」or「こちら」, simply meaning "here".
「あっちゃ」:「あっち」、「あちら」, "over there". Now they should make sense in the context.
1
名【な】 is "name", and Xという名の～ means "～ named X". This is a set expression.
Xという名のならず者
a rogue named X
名の人 doesn't make sense. 名のある人 (or 名がある人) is another set phrase meaning "famous person".
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