Thanks to all the comments, google and J-J dictionaries, I think I've figured it out.
てやる is used here to express a strong will to do the action of the verb
てやろう＝hortative form of てやる
So to connect the pieces, やってやろう means "Let's do it!"
じゃねえか＝じゃないか (/ee/ < /ai/ transformation in informal usage)
じゃないか is used to ask for confirmation ...
いえ is definitely used informally for いいえ, though it's not super casual. More casual options include いや, ううん, or 違う. That being said, you don't necessarily need a word that means 'no'; you can reply to 「フランスに行ったことある？」with just 「ないよ」
You can just enclose the action with parentheses. For example, "Looking up It will be raining soon" will be （見上げながら）雨が降りそうだ。Note that full-width parentheses（）looks nicer than normal parentheses () in Japanese sentence.
Firstly, is there supposed to be a ろ after the end of 住んでるとこ(as in, ところ, and was it cut short due to conversation?)
It was indeed shortened from ところ, but とこ is an extremely common abbreviation of ところ used in colloquial speech, so I wouldn't say that it was "supposed to be" ところ. This is mentioned briefly here (for an entirely different usage of ところ)...
Is it common to censor or soften 馬鹿 as ばしゃ?
No, it is not. As you can see in this page, ばしゃあぁぁ is an original word used by the particular character in that anime, created by the author. It's not used by or known to ordinary people in real life.
It is similar to "right ?"
Used by men
The equivalent used by women would be もんね
It's cold today, right ?