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."
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?"
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:
The following is odd because surely it's easy to go back and just switch it off.
The following is thus natural again: