"진짜 사장 이 자식 완전 재수 똥떵어리지 않냐?" what's the meaning of 똥떵 어리다? and 재수 here?

똥떵어리 means a chunk of excrement.

And there is an English word that just matchs with 똥떵어리.
That is "<em>piece of shit</em>"
재수 came from "재수없다", which basically means "not lucky"

"오늘 재수가 없네" means "I'm unlucky today"

But this is also used to insult and cuss against someone.

Indicating someone "그는 정말 재수없어" means "He is so hateful!"

Anyway, in this case "재수" means, I would say "asshole"

"지금 불난 집에서 인증샷 찍어요?" Does it have the same meaning with "불난 데 부채질하다"?

You got it right.

인증샷을 찍다 is taking a picture of something or oneself against somewhere so that their family and friends or someone else see it.

Korean people tend to do this a lot.

Imagine when your house is burning and there are some tourists doing nothing but

watching it and trying to take a picture of themselves against the burning house.

So almost same meaning as they have each other.

"아까부터 쭉 지켜봤는데 화가 나서 그냥 있을 수가 있어야죠."

화가 나서 그냥 있을 수가 있어야죠 = 화가 나서 그냥 있을 수가 없군요
"I looked at you for some time, and I can't help but intervene."
~ 해서 ~할 수가 있어야죠 indicates anger, annoying. This phrase is used when poeple try to complain to someone.
[for example] 시끄러워서 공부를 할 수가 있어야죠 = 시끄러워서 공부를 할 수가 없잖아요

"형은 머릴 키 커 보일라고 달고 다니냐?" i have no idea about this sentence T T

First, this is sarcastic expression.
"머리를 달고 다니다" 머리를 액세서리에 비유한 것입니다.
"키가 커 보이기 위해서" 

영어로 표현하면 이런 느낌이겠죠? : Your head is only for making yourself look a little more tall, huh?
Your head is only meant to make yourself look more tall?
It basically means "형은 정말 생각없이 행동하군요"

"사장이 논다고 같이 놀 수야 있나요. 월급 받는 처지에" why there is "야" here. i only know ㄹ 수 있다/없다.

-야 in this indicates "how dare I"
In other words, "놀 수야 있나요" = "감히 내가 놀 수 있나요"

"너 시간 나면 저 현수막 좀 확 불 싸질러 버려" i have no idea about this, too

"When you are free, burn that banner out!"

불 싸지르다 = 불지르다

확 : 불길이 세차게 일어나는 모양

"설사 있다 해도 우영 오빨 누가 이겨먹어요" 이겨먹다's meaning is?

이겨먹다 means here "defy"
Example: 선생님을 감히 이겨먹으려 해? (How dare you even defy your teacher!)