What is the difference between "sick" and "sicky"?

Why does he say "a shop", not "this shop"?

He says "a shop" here instead of "this shop" because he is speaking generally. Even though he is referring to the guy closing up this specific shop, he is speaking generally towards the situation of someone closing up a shop in general.

Does "yak something down" mean "vomit something"? What does it mean exactly?

You are correct. To "yak" is a slang expression meaning to vomit, puke, or throw up. In this context, the speaker has used it to say that someone puked/vomited all down the produce aisle. Ex: Peter got food poisoning and yakked all over the table. Ex: She smelt something unpleasant and immediately started yakking.

What's the difference between "tug" and "pull"?

"Tug" and "pull" are very similar and often interchangeable, however there is a slight difference in nuance. To "tug" usually means to gently pull on something, or pull very softly. To "pull" is usually a little bit stronger and refers to using more strength to pull something. Ex: The dog is tugging on the rope. Ex: The horses pull the sleigh.

How many does "two-two pieces" indicate?

Rachel is making a joke here! The girl is wearing a ballerina tutu, which is pronounced like "two-two." Thus, she says "two-two pieces of candy" as a reference to her costume! Ex: Her ballerina tutu was pink. Ex: Is that a tutu you're wearing?

Why is it so important to be the one who announces the heist?

Great question! In the show, the "heist" is a very significant Halloween tradition for them. Due to its significance and having won the heist before, they each think it's their rightful place to announce the heist. I also think announcing the heist is a kind of preemptive action and hope that they'll win the heist. It shows that the competition has already begun before it's even started and that they're all really competitive!